2 Notation and preliminaries
Throughout this paper, let p be a prime and let k be an algebraically closed field of characteristic p. Let G be a finite group such that p∣|G|, and let u be a p-element of G.
For specific groups, we write Altn for the alternating group of degree n, to distinguish it from the algebraic group of type A, and write Symn for consistency. Similarly, a cyclic group of order n will be denoted Zn rather than Cn. Groups of Lie type are given their standard names of SL, PSL, PSp, and so on.
All modules considered are finite-dimensional and defined over k. We denote the trivial module by k or kG if the group needs to be emphasized, and if H is a subgroup of G and M is a kG-module, then M↓H is the restriction of M to H. As usual, ⊕ and ⊗ denote direct sum and tensor product, with Λi(M) and Si(M) denoting the exterior and symmetric powers of M.
If M is a kG-module and u is a p-element, then the action of u on M is conjugate in GL(M) to a triangular matrix and has a Jordan normal form, made up of blocks of various sizes. If the action of u is conjugate to a triangular matrix with Jordan blocks of sizes m1,…,mr, then we say that uhas type(m1,…,mr) on M. We often place the mi in weakly decreasing order, but this is not necessary.
The modules of interest are as follows.
Definition 3.
Let G be a finite group and let k be an algebraically closed field of characteristic p>0. If u is a p-element of G and M is a kG-module, then uacts minimally actively on M if, in the Jordan normal form of u on M, there is at most one Jordan block of size greater than 1, i.e., if u has type (m,1,…,1) for some m≥1 on M, or equivalently if [M,u]∩CM(u) is at most 1-dimensional. We say that M is minimally active if there exists a non-trivial 2-element acting minimally actively on M.
Notice that the identity acts minimally actively on all modules, and all p-elements act minimally actively on 1-dimensional modules.
We use the term minimally active here, following [3], rather than almost cyclic, following [4, 30], because almost cyclic elements need not be p-elements, where p is the characteristic of the underlying field, i.e., unipotent elements of the corresponding GL(M). Since we definitely require this extra hypothesis, we prefer to use this more specific term, to avoid leading the reader to believe we have classified all irreducible groups containing almost cyclic elements.
In our work we often need to know how many conjugates of a given element u generate the normal closure of 〈u〉 in a given group, so we introduce some notation, following [10].
Definition 4.
Let G be a finite group and let u be an element of G. We denote by α(u) the smallest number of conjugates u1,…,uα(u) of u such that
〈u1,…,uα(u)〉=〈uG〉,
i.e., the fewest number of conjugates of u needed to generate the normal closure of 〈u〉. Write
α(G)=maxu∈Gα(u).
Of course, if a group has even order and is not dihedral, then α(G)≥3, and α(G) is the maximum of α(u) for all elements u of prime order. In [10], various bounds for almost simple groups were obtained, and we will use them frequently to get general constraints on finite groups with elements acting minimally actively on a simple module. For example, in Lemmas 11 and 12 we show that α(u)=2 for some specific conjugacy classes of permutations inside symmetric groups.
We collect several basic facts about minimally active modules now.
Lemma 5.
Let G be a finite group and let M be a faithful kG-module.
If u acts minimally actively on M, then u acts minimally actively on any submodule or quotient of M, and on the dual of M.
If u is contained in a subgroup H of G and acts minimally actively on M, then u acts minimally actively on M↓H.
If M=M1⊕M2 and u acts minimally actively on M, then 〈uG〉 acts trivially on at least one of the Mi.
If M is simple and u acts minimally actively on M, then
dim(M)≤α(u)⋅(o(u)-1).
More generally, if a=dim(M)-dim(CM(u)), then dim(M)≤a⋅α(u).
Suppose that M and N are kG-modules with dim(M)≤dim(N), and such that u acts non-trivially on M⊗N. We have that u acts minimally actively on M⊗N if and only if either dim(M)=1 and u acts minimally actively on N, or p is odd, dim(M)=dim(N)=2, and u acts non-trivially (i.e., has type (2)) on both M and N.
If u acts non-trivially and minimally actively on Λ2(M), then u has type (2),
(2,1) or (3) on M, or p is odd and u has type (4) on M.
If u acts non-trivially and minimally actively on S2(M), then p is odd and u has type (2) on M, or p≥5 and u has type (3) on M.
Proof.
The first three parts are clear. For the fourth part, note that the codimension a of CM(u) is at most o(u)-1, whence the codimension of CM(〈u,ug2,…,ugr〉) is at most ra≤r(o(u)-1). If G is generated by r conjugates of u, then this is CM(G)=0, so that dim(M)≤ra≤r(o(u)-1), as claimed.
For (v), note that if dim(M)=1 and u acts minimally actively on N, then the result is clear, and if p is odd and u acts as a single Jordan block of size 2 on both M and N, then u acts on M⊗N with type (3,1), so one direction holds. For the other, if dim(M)=dim(N)=2 and p=2, then u acts on M⊗N with type (2,2), and otherwise dim(N)≥3. If u acts trivially on M, then it must act non-trivially on N, and the action of u on M⊗N contains two copies of the action of u on N (as dim(M)≥2) so that u cannot act minimally actively. If u acts non-trivially on M and dim(N)≥3, then M contains a u-invariant subspace on which u acts with type (2), and N contains a u-invariant subspace with type either (3) or (2,1). In the first case, u acts on the tensor product of these subspaces as (4,2) (or (3,3) if p=3), and in the second as (3,2,1) (or (2,2,2) if p=2), so u does not act minimally actively in either case, by applying (i).
For the statements about exterior and symmetric powers, recall that
S2(A⊕B)=S2(A)⊕(A⊗B)⊕S2(B)
and similarly for exterior squares. Thus if u has at least three blocks, then it contains a submatrix of type (2,1,1), and the symmetric and exterior squares of this have two blocks of size 2.
Table 1
Types of symmetric and exterior squares.
| Type | Symmetric square | Exterior square |
| (2) | (3) (p≠2), (2,1) (p=2) | (1) (all p) |
| (2,1) | (3,2,1) (p≠2), (2,2,1,1) (p=2) | (2,1) (all p) |
| (3) | (5,1) (p≠2,3), (3,3) (p=3), (4,2) (p=2) | (3) (all p) |
| (3,1) | (5,3,1,1) (p≠2,3), (3,3,3,1) (p=3), (4,3,2,1) (p=2) | (3,3) (all p) |
| (4) | (7,3) (p≠2,5), (5,5) (p=5), (4,4,2) (p=2) | (5,1) (p≠2), (4,2) (p=2) |
| (5) | (9,5,1) (p≠2,5), (5,5,5) (p=5), (8,4,3) (p=2) | (7,3) (p≠5), (5,5) (p=5) |
From Table 1, we see that u acts minimally on Λ2(M) and S2(M) when claimed, and that u cannot act minimally on either of these when (3,1) or (5) is a submatrix of the type of u on M. All other possibilities are in Table 1, and this completes the proof.
∎
In characteristic 2, we will have to consider modules that are not exterior squares, but exterior squares with one or two trivial composition factors removed.
Lemma 6.
Let p=2, let G be a finite group and let M be a faithful, simple module of dimension at least 6. If V is obtained from Λ2(M) by removing at most two trivial composition factors, then V is not minimally active for any non-trivial 2-element of G.
Proof.
The exterior square of a block of size 6 has type (8,6,1), so even a submodule of codimension 2 cannot be minimally active for u. Similarly, the exterior square of a matrix of type (4,1) has type (4,4,2), so again we cannot find a minimally active submodule of codimension 2 for u. The exterior square of a matrix of type (3,1,1) has type (3,3,3,1) and that of (2,14) has type (24,17), so again this cannot work. Every type for u acting on M contains one of these types as a submodule, hence u cannot act minimally actively on V.
∎
We now give a lemma on when a power of an element can be minimally active. This uses the classification of groups generated by transvections given in [17].
Lemma 7.
Let G be a finite group and let M be a faithful, simple kG-module. Let u be a p-element and suppose that a non-trivial element v of 〈up〉 acts minimally actively on M.
Then v acts as a transvection on M. Furthermore, G contains a classical group in its natural representation as a normal subgroup, or p=2, k contains F4, and G is either 3⋅Alt6≤GL3(k) or 3⋅PSU4(3)≤GL6(k).
Proof.
The pth power of a single Jordan block of size ap is the sum of p blocks of size a; from this it is easy to see that the pth power of a single block of size ap+b is the sum of b blocks of size a+1 and p-b blocks of size a. In order for this to be minimally active, we must have a=b=1. Thus u is the sum of one block of size pa+1 for some a and blocks of size at most pa, and an element v of order p in 〈u〉 is a transvection, i.e., has type (2,1n-2) for n=dim(M).
Since G possesses a faithful simple module in characteristic p, Op(G)=1. (There are many ways to see this: one is that parabolic subgroups act reducibly in the general linear group, and the normalizer of a p-subgroup is contained in a parabolic.) The subgroup H generated by all conjugates of v is a normal subgroup of G, whence acts semisimply on M as a sum of conjugate modules but also v acts minimally actively, whence H acts irreducibly on M by Lemma 5
(iii). Thus H is an irreducible subgroup of GL(M), with Op(H)=1, containing a transvection, so is one of the groups on Kantor’s list in [17, Theorem II].
Of these, we need to check which have a transvection as a proper power of a p-element. Classical groups certainly do (cases (T1) and (T2) in Kantor’s list), whereas no 2-element powers to a transposition in Symn (cases (T3) and (T9)), and case (T6) has Sylow p-subgroups of exponent p. Cases (T4) and (T8) are not irreducible, and (T5) and (T7) have a single class of involutions, which must be transpositions, and do not have exponent 2, so are examples. This exhausts the list.
∎
If the Sylow p-subgroup of G is cyclic, then we can say more about minimally active modules. This next lemma is a generalization of [3, Propositions 3.7 and 3.9], and the proof follows the same method. We do not give all the background on Green correspondence needed for their proof here, and instead refer to [3, Section 3] and the references therein.
Lemma 8.
Let G be a finite group and let M be a faithful, simple kG-module. Suppose that the Sylow p-subgroup U of G is cyclic and generated by u.
If NG(U)/U is abelian (for example, if CG(u)=Z(G)⋅〈u〉), then u acts minimally actively on M if and only if dim(M)≤o(u)+1.
If CG(u) is abelian and M is minimally active, then dim(M)<2⋅o(u).
If M is minimally active, then dim(M)≤o(u)+b, where b<|CG(u)|.
Proof.
By Lemma 7 we may assume that if M is minimally active, that it is u that acts minimally actively.
If dim(M)=a⋅o(u) for some integer a≥1, then M is projective and u acts with a blocks of size o(u); thus M is minimally active if and only if dim(M)=o(u). Hence we can suppose that M is not projective. Let V denote the Green correspondent of M in NG(U), so that M↓NG(U)=V⊕X, where V is an indecomposable kNG(U)-module and X is a relatively 〈up〉-projective kNG(U)-module.
We next aim to understand the action of u on V. Note that V is indecomposable and all indecomposable modules for p-soluble groups are uniserial, and all composition factors of V lie in the same block of NG(U) and have the same dimension m. (This follows since the Brauer tree of a block of a p-soluble group is a star.) Thus the action of u is as m blocks of the same size r, where dim(V)=mr.
Suppose that X=0. Since u acts non-trivially on M, this means that r>1, so that m=1 if and only if u acts minimally actively on M. In particular, if M is minimally active, then dim(M)<o(u). Since m is the dimension of a simple NG(U)/U-module, if NG(U)/U is abelian, then this means m=1, so if X=0, then u acts minimally actively on M. This proves all parts of the result when X=0.
Since X≠0, and clearly u acts non-trivially on X, it must act trivially on V by Lemma 5
(iii), so V is an indecomposable kNG(U)/U-module, hence a simple module as this is a p′-group. Since X is relatively 〈up〉-projective, by Green’s indecomposability criterion we see that all Jordan blocks of u on X have size a multiple of p, so there is exactly one, and dim(X)≤o(u). This completes the proof of (i) since dim(V)=1 in this case.
Otherwise we need to bound the dimension of a simple NG(U)/U-module: since |NG(U)/CG(U)| has order dividing p-1, if CG(U) is abelian, then any simple NG(U)-module has dimension at most p-1: thus
dim(M)≤o(u)+(p-1)<2o(u),
as needed. Finally, as NG(U)-modules are orbits of CG(U)-modules by Clifford’s theorem, they have dimension at most |CG(U)|-1 (the ‘-1’ is because the trivial is always in a separate orbit) proving the third part.
∎
We move on to examining p-elements acting on direct sum and tensor product decompositions.
Lemma 9.
Let G be a finite group and let u∈G be a p-element. Suppose that u acts minimally actively on a faithful, simple kG-module M. Suppose that H is a normal subgroup of G and that G=H〈u〉. Let 1<t=|G:H| and suppose that M↓H is the sum of t non-isomorphic simple modules. The action of u on M, and of ut on each composition factor of M↓H, is as a single Jordan block, of size the dimension of the module.
Conversely, if ut acts on each composition factor of M↓H as a single Jordan block, then u acts minimally actively on M with a single Jordan block.
Proof.
We note at the start that t is a power of p. Since the restriction of M to H is the sum of t non-isomorphic modules, we have the decomposition
M↓H=M1⊕M2⊕⋯⊕Mt,
where Mi⋅u=Mi+1. Suppose that m=(m1,…,mt) is a fixed point of u, so that in particular each mi∈Mi is a fixed point of ut. Note that, since m⋅u=m, we must have that mi⋅u=mi+1, whence there is a one-to-one correspondence between the u-fixed points of M and the ut-fixed points of M1, and in particular their dimensions are equal.
Writing d=dim(M1), so that dim(M)=dt, if u has type (a,1dt-a), then dim(M)〈u〉=dt-a+1. This has to be equal to dim(M1)〈ut〉, which is at most d. This yields
dt-a+1≤d.
First suppose that a≤t, so that ut=1. This yields
dt-t+1≤dt-a+1=d,
i.e., (d-1)(t-1)≤0, yielding either d=1 or t=1, the latter of which is impossible.
Thus suppose that ut≠1. Since u has type (a,1dt-a), we need to know how ut acts: a block of size a, when raised to the tth power, has type ((α+1)β,αt-β), where a=tα+β and 0≤β<t. Thus the action of ut on M has type
((α+1)β,αt-β,1td-a).
These must be distributed equally among the t distinct Mi, whence t∣β and this means that β=0. This means that ut acts on M with type (tα,1t(d-α)), which means that it acts on each Mi with type (α,1d-α).
Thus we now have that
1+t(d-α)=dim(M)〈u〉=dim(M1)〈ut〉=d-α+1,
so t=1 (again, impossible) or d=α, in other words, ut acts with a single Jordan block, and therefore so does u, as claimed.
For the converse, since ut fixes a unique 1-space on each Mi, any fixed point of u must lie inside this span. But ut acts on this t-space as a transitive permutation module, hence fixes a unique 1-space. Thus u acts with a single Jordan block, as claimed.
∎
Along with unipotent elements permuting direct sums, we need unipotent elements permuting tensor products.
Lemma 10.
Let G be a finite group and let u be a p-element. Let H be a normal subgroup of G such that G=H〈u〉, and let M be a faithful, simple kG-module that is not isomorphic to a non-trivial tensor product of two modules, and whose restriction to H factors as a tensor product
M1⊗M2⊗⋯⊗Mt
of kH-modules, where |G:H|=t>1 and dim(Mi)>1. If u acts minimally actively on M, then one of the following holds:
p=t=2,
dim(Mi)=2,3,
u2 has type (2,2) or (4,4,1) on M.
p=t=3,
dim(Mi)=2,
u3 has type (3,3,2) on M.
Conversely, if p, t, dim(M) and the action of ut is as above, then u acts minimally actively on M.
Proof.
First suppose that ut=1. Notice that for any element v1 in M1, writing vi+1=vi⋅u, we can arrange the Mi so that vi∈Mi and v1⊗⋯⊗vt is fixed by u. The subspace spanned by all other monomials in the tensor product is also fixed by u, so M↓〈u〉 is the sum of a trivial module of dimension dim(Mi) and a permutation module with basis the monomials in the tensor product. Since all other orbits than v1⊗⋯⊗vr have length greater than 1, if u is minimally active, then there is a single orbit on the monomials. However, this is clearly impossible, for example since the number of monomials is dim(Mi)t-dim(Mi)>t, unless p=t=dim(Mi)=2.
We therefore may assume that 〈u〉∩H≠1, so that ut is a non-trivial p-element. If u acts minimally actively on M, then ut has at most t non-trivial blocks. We will prove that, for almost all possible Jordan normal forms of ut, there must be more than t non-trivial blocks in its t-fold tensor power. Note that, if this is shown for a block of type (α1,…,αr), then it is shown for any type (β1,…,βs) with s≥r and αi≤βi for all 1≤i≤r, because for the cyclic group of order a, the kZa-module with indecomposable summands of dimensions α1,…,αr is a submodule of that with dimensions β1,…,βs, and hence the t-fold tensor power of the former is also a submodule of the latter.
Suppose that p=2. If ut is a single block of size 2, then the t-fold tensor power of the action of ut is as 2t-1 blocks of size 2, and this is greater than t for t≥3. Thus if t≥4 (as t must be a power of 2), then u cannot act minimally actively at all. Thus we may assume that t=2.
If ut is a block of size 3, then the tensor square has type (4,4,1), so is a candidate. If ut is a block of size 4, then the tensor square has type (4,4,4,4), so we eliminate all blocks of size at least 4, leading to the result in the lemma.
We now check that these two cases occur. For dim(M1)=2, we have that u has order 4 and its square acts as (2,2), so u must have a single block of size 4. For dim(M1)=3, u has order 4 and its square acts as (4,4,1), so u must act as (8,1) (as blocks of sizes 5, 6 and 7 square to have types (3,2), (3,3) and (4,3), respectively). Thus u must be minimally active in these cases.
Now suppose that p is odd, and again we consider the t-fold tensor power of a block V of size 2. For the first few tensor powers, we describe them now. In this table we assume that t<p.
| t | type |
| 1 | (2) |
| 2 | (3,1) |
| 3 | (4,22) |
| 4 | (5,33,12) |
| 5 | (6,44,25) |
| 6 | (7,55,39,15) |
These are easily generated as tensoring a block of size i by a block of size 2 yields two blocks, of size i-1 and i+1, at least when i<p. It is easy to see that the start of the t-fold product for arbitrary t is
(t+1,(t-1)t-1,(t-3)(t-1)(t-2)/2-1,…),
and so for t≥5 there are more than t non-trivial blocks of size less than t in the (t-1)-fold tensor power of a single block. This means that there are more than p non-trivial blocks in the p-fold power of a single block, and at least two of them have size p, for p≥5. (If p=3, tensoring (3,1) by (2) yields (3,3,2), so the second statement holds but not the first.)
For t>p, we write this as a single tensor product of V⊗p and V⊗(t-p). The first of these contains two blocks of size p, whose product with V⊗(t-p) consists entirely of blocks of size p, and hence the product contains at least 2(t-p)>t blocks of size p when t>p is a power of an odd p.
This proves that u is not minimally active if p≥5, or p=3 and t≥9. If p=t=3 and V is a block of size 3, then V⊗3 is the sum of nine blocks of size 3, and if V is the sum of a 1- and 2-dimensional module, then V⊗3 has type (35,22,14). Thus if dim(Mi)≥3, then we are also done.
We are left with V having dimension 2, in which case V⊗3 has type (3,3,2), as we saw above. If u has order 9 and u3 acts as (3,3,2), then we use the table below that displays the blocks of u3, given a block of u.
| u | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| u3 | 1 | 12 | 13 | 2,12 | 22,1 | 23 | 3,22 | 32,2 |
We clearly see that u cannot have blocks of size other than 8, and hence u acts with a single block, as claimed.
∎
The next two lemmas are needed in our analysis of minimally active modules for (central extensions of) symmetric groups.
Lemma 11.
Let G=Symn for some n, and suppose that u∈G is of order at least 4 and has no cycles of length 1 or 2. Then α(u)=2.
Proof.
Of course, u=(1,2,…,n) and v=(1,2) generate Symn, so 〈u,uv〉 has index at most 2 in Symn, and we see that α(u)=2 when u is a single cycle.
Suppose that u has cycle type (m1,…,mr), with all mi≥3, and mr≥4. Write n0=0, ni=∑j=1imi, and for 1≤i≤r-1, let
σi=(ni-1-(i-2),ni-1-(i-3),…,ni-i,n-i),
a cycle of length mi. Finally, let
σr=(nr-1-(r-2),nr-1-(r-3),…,n-r-1,n,n-r),
and let u be the product of the σi. The second generator is
v=(n1,n-1)(n2-1,n-2)…(nr-1-(r-2),n-(r-1))(n-r,n).
Notice that uv is just the (n-1)-cycle (1,…,n-1), and that
[u,v]=(1,n1+1)(n1,n2,n-1)(n2-1,n3-1,n-2)
…(nr-2-(r-3),nr-1-(r-3),n-(r-2))
(n-r,n-(r-1),nr-1-(r-2),n),
so that [u,v]6 is a double transposition. Letting H=〈u,v〉, we note that H is transitive and contains an (n-1)-cycle, hence 2-transitive and so primitive. Since it contains a double transposition, and by [6, Example 3.3.1] a primitive subgroup of Symn containing a double transposition contains Altn for n≥9, we get α(u)=2 in this case as well.
The remaining cases to check are for n=7,8 and u with cycle type (4,3), (5,3) and (4,4). In the first case, Sym7 is generated by (1,2,3,4)(5,6,7) and (1,2,3,5)(4,6,7), and in the second and third cases, the group Alt8 is generated by (1,2,3,4,5)(6,7,8) and (1,2,3,4,6)(5,7,8), and also by (1,2,3,4)(5,6,7,8) and (1,2,5,6)(4,3,7,8).
∎
Lemma 12.
Let G=Symn for some n≥9. If u∈G has cycle type (n-2,2), (n-4,2,2) or (n-6,2,2,2), then α(u)=2. If n=10 and u has cycle type (4,4,2), then α(u)=2 also.
Proof.
In the first case, let
u=(1,n-1)(2,3,…,n-3,n,n-2),v=(2,n-1)(n-2,n).
Again, uv=(1,2,…,n-1) and the same proof applies as Lemma 11, as v is itself a double transposition.
In the second case, let
u=(1,n-1)(2,n-2)(3,…,n-4,n,n-3),v=(2,n-1)(3,n-2)(n-3,n).
Then uv is as in the previous case, but now we need to find an element of small support, and this is
[u,v]=(1,3,n,n-3,n-2)(2,4,n-1),
and so [u,v]5 is a 3-cycle, and we are done.
In the third case, let
u=(1,n-1)(2,n-2)(3,n-3)(4,…,n-5,n,n-4),v=(2,n-1)(3,n-2)(4,n-3)(n-4,n).
Again, uv is as before, but if n≥10, then
[u,v]=(1,3,5,n-2)(2,4,n,n-4,n-3,n-1),
and [u,v]6=(1,5)(3,n-2), as needed. If n=9, then u2 is a 3-cycle, and we are again done.
Finally, we simply give generators of Sym10 of the appropriate cycle types:
(1,2,3,4)(5,6,7,8)(9,10) and (1,5,4,7)(2,3,8,10)(6,9).
This completes the proof.
∎
We also need to determine better bounds on α(u) for u a unipotent element in GLn(2) than α(u)≤n given in [10]. While this bound is sharp for transvections, simply by considering the fixed-point subspace, we need elements close to regular elements. Indeed, by [8], with the exception of SL4(2), SLn(p) is generated by two regular unipotent elements for all primes p, and all n≥3. We give this in a lemma for reference.
Lemma 13 ([8]).
Let G=SLn(p) for some n≥2. If u is a regular unipotent element of G, then α(u)=2.
Proof.
This is proved in [8] for all cases except for SL4(2)=Alt8, where the regular unipotent class is in bijection with the class containing u=(1,2,3,4)(5,6). Letting v=(1,5,7,8)(4,6), we note that 〈u,v〉 generates a primitive subgroup of Altn containing (uv2)5=(5,8,6). This completes the proof.
∎
From this, we can get that if u is a 2-element of maximal order in SLn(2), then α(u)≤4; it is likely that this could be improved still further, but not without considerably more work.
Lemma 14.
Let G=SLn(2), and let u be a unipotent element of maximal order in G. If n is even, then α(u)≤3 and if n is odd then α(u)≤4.
Proof.
Write V for the natural module for G. Suppose that u has type (a,1n-a) on V for some a≥n2+1, so that in particular CV(u) has dimension less than 12dim(V). Let u1 and u2 be regular unipotent generators of SLa(2), written as matrices in G, so with type (a,1n-a). The action of H1=〈u1,u2〉 on V has a single simple submodule W of dimension a, and all other simple submodules trivial.
Write v1,…,va for a basis of W, and extend the basis to va+1,…,vn on which H1 acts trivially. Let u3 act as follows:
vi⋅u3={v1+vn,i=1,vi,2≤i≤n-a+1,vi+vi-1,n-a+2≤i≤n.
Of course, u3 has the correct type. We claim that H=〈u1,u2,u3〉 acts irreducibly on V, so let X denote an H-submodule of V. Since W is a simple H1-submodule, either X∩W=0 or W≤X: if W≤X, then v1∈X, so vn∈X and we see that each vi∈X, so that V=X. Thus X∩W=0, and so H1 acts trivially on X, yielding X is a subspace of 〈va+1,…,vn〉. However, repeated application of u3 to any element of this space eventually leaves it, as we must project onto va, so that X=0. Thus H is irreducible on V, containing a copy of SLa(2) acting on V in a non-self-dual way, hence H≰Sp(V). Since H contains a transvection, we can apply [17, Theorem II]: either H=G, H is a classical group (all contained in Sp(V) as the characteristic is 2) or a symmetric group (again, contained in Sp(V) as the simple modules are self-dual), so since H≰Sp(V), we have that H=G, as needed.
If n-a+1=a, i.e., n=2a+1, then the above argument fails: in this case, generate SLn-1(2) with three elements, and use the fourth to get the full SLn(2).
If the ui have another type, with a single block of size a and various smaller blocks instead, then choose the ui exactly as before: note that every subspace of V stabilized by ui is also stabilized by the previous ui, and so since we had an irreducible subgroup before we must have an irreducible subgroup again. Since it still contains a transposition, we still have SLn(2), as needed.
∎
We end this section by giving the notation used for almost quasisimple groups. Our groups G will have the property that G=〈F*(G),u〉 for some p-element u, that G0=F*(G) is quasisimple and that Z(G)=Z(G0). If M is a faithful simple kG-module then this yields an embedding of G into GL(M). Our conditions on G are equivalent to the image H of G in PGL(M) being almost simple, and H′ being simple with H/H′ generated by a p-element of H.
3 Reduction to almost simple groups
This section uses Aschbacher’s classification of maximal subgroups of classical groups [1] (see also [19, 25, 31], and in particular [19], which modifies the classes of Aschbacher, and whose notation we will use here) to reduce to the case given at the end of the last section, where G is an almost quasisimple group. Thus we have eight classes 𝒞1,…,𝒞8 of maximal subgroups, together with almost quasisimple groups 𝒮. We will determine which elements of the 𝒞i contain minimally active elements.
We assume in this section that G is a subgroup of GL(M) for an n-dimensional k-vector space M, with u∈G being a p-element acting minimally actively on M. As we are only concerned with irreducible modules, we stipulate that G acts irreducibly on M. In particular, G cannot lie in a parabolic subgroup, class 𝒞1.
If G acts imprimitively on M, then G stabilizes a direct sum decomposition
M=M1⊕M2⊕⋯⊕Mt
of M. Taking this decomposition to be as fine as possible, we see that G is a subgroup of GLn/t(k)≀Symt (Aschbacher’s class 𝒞2). The action of u on this group is in Lemma 9: u must act with a single non-trivial block, t is a power of p, and ut acts with a single block on each Mi. Hence G lies in a wreath product A≀B, where ut∈A acts on M1 with a single Jordan block, and the t-cycle (1,…,t) lies in B. Furthermore, given such a setup we always obtain an element u. Thus there are many groups acting imprimitively on M with minimally active elements.
If we extend the field and the module is no longer irreducible (G is contained in an extension field subgroup, class 𝒞3), then we can apply Lemma 5
(iii) to see that G cannot have a minimally active element.
Suppose that G acts primitively and absolutely irreducibly on M, and preserves a tensor decomposition
M=M1⊗M2⊗⋯⊗Mt,
so that G is a subgroup of GLm(k)≀Symt with n=mt. Now we can apply Lemmas 5
(v) and 10, which show that one of the following holds:
n=4 and p is odd, with G a subgroup of SL2(k)×SL2(k) and o(u)=p acting with type (3,1),
n=4 and p=2, with o(u)=4 acting with type (4),
n=8 and p=3, with o(u)=9 acting with type (8),
n=9 and p=2, with o(u)=8 acting with type (8,1).
We also showed in that lemma that these cases occur, and we will not comment further on this case. These are classes 𝒞4 and 𝒞7.
If G is contained inside a subfield subgroup in the class 𝒞5 (of the form NGLn(k)(GLn(k0))), then since our property is independent of the field over which we take our module M, we replace k by k0 and so this case can be ignored.
If G is contained inside another classical group (i.e., class 𝒞8), then we apply this classification of maximal subgroups to that group instead.
Thus we are left with 𝒞6, extraspecial-type subgroups, and 𝒮, which is the focus of all subsequent sections of the paper.
Let r≠p be a prime, and let R denote an extraspecial group of order r1+2m for some m. If r=2, we allow R to be either an extraspecial group or the central product with Z4. Note that a faithful, irreducible representation of R in characteristic not r has dimension n=rm, so we may embed R into GLn(k) for k an algebraically closed field of characteristic p, and k any field of characteristic p except when R=Z4∘r1+2m and k needs a fourth root of unity. Let G denote the normalizer in GLn(k) of R, and let M be the natural module for G. More information about G can be found in [19, 25, 31].
We want to prove that if G contains a minimally active element, then the parameters r,p,m are very tightly controlled. To do so, we need to know something about G/R, which is a classical group. In Section 7, particularly Proposition 26, we get information about the orders of p-elements of classical groups in characteristic different from p. Rather than deferring the proof of this result until then, we include it here, but use the definitions and notation from that section. The reader is recommended to skip the proof of this result until they have reached Section 7; the proof is similar to those contained in Section 8.
Proposition 15.
If u∈G acts minimally actively on M, then r=2 and one of the following holds:
m is an odd prime, p=2m-1 is a Mersenne prime, and u has order p, acting with type (p,1),
m is a power of
2,
p=2m+1 is a Fermat prime, and u has order p, acting with type (p-1),
m=2,
p=3,
o(u)=3 acting with type (3,1) (not all elements of order
3
have this property),
m=3,
p=3,
o(u)=9, acting with type (8).
Proof.
If r is odd, then Op′(G)=R⋊Sp2m(r) is a split extension, and so if u is a p-element of G, we may assume that u lies in H=Sp2m(r). By [9, Section 5], if p is odd, then H acts on M as the direct sum of the two Weil modules, of dimensions 12(rm-1) and 12(rm+1), and so u cannot act minimally actively on M by Lemma 5
(iii). If p=2, then M is uniserial of length 3 (see [9, Lemma 5.2]) with socle series W, k and W, where W is a Weil module, and since W is not minimally active for p=2 by Theorem 28 we eliminate this case as well.
Thus r=2 and hence p is odd. Here G/R≅H, where H=GO2m±(2) or Sp2m(2), and the orthogonal-type groups are contained in R.Sp2m(2), so we will work solely with that group. Let d denote the order of 2 modulo p, so that p divides Φd(2). We have that α(u)≤m+3 in all cases, since p is odd.
Let us suppose that the image in H of u lies in a Levi subgroup of H, say Sp2m-2a(2)×Sp2a(2). Taking preimages in G yields a central product G1∘G2, where
G1=R1.H1=(4∘22(n-a)+1).Sp2(m-a)(2),G2=R2.H2=(4∘22a+1).Sp2a(2).
The action of this group on M is a tensor product of actions of the Gi, and since the tensor product of two modules cannot be minimally active unless they both have dimension at most 2 by Lemma 5
(v), we have that m=2 and p=3. This case will be considered later.
Thus we may assume that the Sylow p-subgroup of G, and hence H, does not lie in any proper Levi subgroup of H. This in particular means that d divides 2m.
Suppose that d is either 2m, or m is odd and d=m; in both cases d is regular, so CH(u) is abelian and of odd order since the image of u is semisimple in H and therefore CH(u) is reductive. Thus CG(u) splits as the direct product of CR(u) and a subgroup C isomorphic to CH(u). Thus R=CR(u)∘[R,u], and since u does not lie in a Levi subgroup of H, CR(u)=Z(R)≤Z(GL(M)).
This shows that CG(u)=Z(R)×C, and in particular CG(u) is abelian, so that 2m=dim(M)≤2o(u) if u acts minimally actively on M by Lemma 8 (ii). As dim(M)=2m and o(u)∣Φd(2)∣(2m±1), it follows that if (2m±1) is not a prime power, then 2o(u)≤23(2m±1)<2m, and so o(u)=2m±1. Thus 2m±1=Φd(2), so that m is either a power of 2 or is a prime, and 2m±1 is a Fermat or Mersenne prime, or is 9=23+1, but 2 has order 2 modulo 3, not 6.
In these cases, CH(u) is abelian and indeed is simply 〈u〉, so the subgroup CG(u)=Z(R)×〈u〉 is cyclic, and we may apply Lemma 8 again to show that u acts minimally actively on M, and we are done.
If d=m for m even, then the Sylow p-subgroup of H has rank 2, and lies inside the Levi subgroup Spm(2)×Spm(2), so u can only act minimally actively if m=1, i.e., p=3, as we saw above.
Suppose that the Sylow p-subgroup of G is abelian, so that o(u) is a divisor of Φd(2), for d∣2m with d≠m,2m. If d is odd, then d≤m2, and if d is even then d≤2m3: in the first case, o(u)≤(2m/2-1) and in the second o(u)≤(2m/3+1). Since α(u)≤m+3 and dim(M)=2m, we get
(m+3)⋅(o(u)-1)≤2m,
which has solutions only for m≤4: noting that d≠1 and d∣2m, we get d=2 for m=3,4. In this case p∣Φ2(2)=3 and the Sylow 3-subgroups of Sp6(2) and Sp8(2) are non-abelian, so there are no solutions.
If the Sylow p-subgroup of G is non-abelian, then d divides m and also pd is at most m if d is odd and 2m if d is even. Suppose that p≥5: then d≥3, and if d=3, then p=7, if d=4, then p=5, and if d≥5, then p≥11. If p=5, then m≥10, and if p≥7, then m≥21. We have d≤2m5, and
o(u)≤m(22m/5-1),
where the Weyl contribution is m and the toral contribution is of course 22m/5-1. Thus the inequality α(u)⋅o(u)>dim(M) for u to act minimally actively becomes
m(m+3)(22m/5-1)≥2m
for m≥10, and the only solutions are for m=10,11,12, so we must have p=5. In this case o(u)≤25 so we get 25(m+3)≤2m for m≥10, and this obviously has no solutions.
We therefore have p=3 so d=2, and o(u)≤3m. This yields the inequality
(3m-1)(m+3)≥2m,
which is satisfied for m≤7. The Sylow 3-subgroup of Sp2m(2) lies inside the Levi subgroup Sp6(2)×Sp2m-6(2) for 4≤m≤7, so these cases need not be considered.
We now collect together the cases we need to check, which are only m=2,3 for p=3. For (Z4∘21+4).Sp4(2), we have that the element u is contained in (Z4∘21+4).(Sp2(2)×Sp2(2)), so the action of u on M can be factored as the tensor product of two matrices. If u lies in one of the factors, then this would have type (22), but if it is diagonal, then it would act as 2⊗2, so type (3,1), minimally active.
For (Z4∘21+6).Sp6(2)≤SL8(9), if o(u)=3, then α(u)≤4 (and hence u cannot act minimally actively) unless u lies inside a Levi subgroup Sp2, which of course means that u is not minimally active (as u cannot lie inside such a Levi).
Thus o(u)=9: inside Z3≀Z3≤Sym9, elements of order 9 square to the fixed-point-free class of elements of order 3, and so inside Sp2(2)≀Sym3≤Sp6(2) we see that elements of order 9 power to the class that lies diagonally across all three Sp2(2) factors. Therefore the action of u3 on M has type (3,3,2), as this is the type of the third tensor power of a block of size 2. As blocks of size 5, 6, 7 and 8 power to have types (2,2,1), (2,2,2), (3,2,2) and (3,3,2), respectively, we see that u must have a single block of size 8, hence is minimally active, as needed.
∎
Thus we have considered all of the 𝒞i, and so we may assume that G is a member of 𝒮. Furthermore, this completes the proof of Theorem 1.
4 Alternating groups
In this section we use the notation at the very end of Section 2: let G0 be a central extension of an alternating group Altn for n≥5, let u be a p-element of G such that G=〈G0,u〉. (Note that, since Out(G0) is a 2-group, we have that u∈G0 unless p=2.) When n=5,6,7,8 we get very different answers to the general case: for n=5 this is because Alt5 is isomorphic to SL2(4) and PSL2(5); for Alt6 it is because of the extra outer automorphism, the exceptional triple cover, and the isomorphism with PSL2(9); for Alt7, it is the exceptional triple cover; and for Alt8 it is the isomorphism with SL4(2).
Because of this, the first proposition deals with those four individual groups. Because of their small order, one can check all calculations easily on a computer, and we just say a few words about its proof. In this proposition, cases (ii) and (iii) are written as if they are general statements, which they are, even though there is only one instance of each in the range 5≤n≤8.
Proposition 16.
Let G0 be a central extension of Altn for 5≤n≤8, and let u be a p-element of G such that G=〈G0,u〉. Let V denote the non-trivial composition factor of the permutation module for G. If u acts minimally actively on a non-trivial simple module M, then (up to outer automorphism in the case n=6) one of the following holds:
G=Altn for p odd and G=Symn for p=2,
u is a single cycle of length pa for some pa≤n and M=V,
G=Altn for n=2a+2,
a≥2,
p=2,
u has cycle type (2a,2) and M=V,
G=Symn for n=2a+4 with a≥2,
p=2,
u has cycle type (2a,2,2) and M=V, with u acting as (2a,1,1),
G=Alt5,
p=2,
o(u)=2 and dim(M)=2,
G=Sym5,
p=2,
o(u)=4 and dim(M)=4,
G=Alt5,
p=3,
o(u)=3 and dim(M)=3,
G=2⋅Alt5,
p=3,
o(u)=3 and dim(M)=2,
G=Alt5,
p=5,
o(u)=5 and dim(M)=5,
G=2⋅Alt5,
p=5,
o(u)=5 and dim(M)=2,4,
G=PGL2(9) or G=M10,
p=2,
o(u)=8 and dim(M)=8 (three and one representation respectively),
G=3⋅M10,
p=2,
o(u)=8 and dim(M)=6,9,
G=3⋅Alt6,
p=2,
o(u)=2,4 and dim(M)=3 (four representations),
G=Alt6,
p=3,
o(u)=3 and dim(M)=3,
G=2⋅Alt6,
p=3,
o(u)=3 and dim(M)=2 (two representations),
G=2⋅Alt6,
p=5,
o(u)=5 and dim(M)=4 (two representations),
G=3⋅Alt6,
p=5,
o(u)=5 and dim(M)=3,6 (two representations each),
G=6⋅Alt6,
p=5,
o(u)=5 and dim(M)=6 (four representations),
G=Alt7,
p=2,
u=(1,2,3,4)(5,6) and dim(M)=4,
G=2⋅Alt7,
p=3,
u is the preimage of the element (1,2,3)(4,5,6) and dim(M)=4,
G=2⋅Alt7,
p=5,
o(u)=5 and dim(M)=4,
G=3⋅Alt7,
p=5,
o(u)=5 and dim(M)=3,6 (two representations each),
G=6⋅Alt7,
p=5,
o(u)=5 and dim(M)=6 (four representations),
G=2⋅Alt7,
p=7,
o(u)=7 and dim(M)=4,
G=3⋅Alt7,
p=7,
o(u)=7 and dim(M)=6 (two representations),
G=6⋅Alt7,
p=7,
o(u)=7 and dim(M)=6 (four representations),
G=Alt8,
p=2,
u=(1,2)(3,4)(5,6)(7,8) and dim(M)=4,
G=Alt8,
p=2,
u=(1,2,3,4)(5,6,7,8) and dim(M)=4, with u acting as (3,1),
G=Alt8,
p=2,
u=(1,2,3,4)(5,6) and dim(M)=4,
G=2⋅Alt8,
p=7,
o(u)=7 and dim(M)=8.
In all cases, unless otherwise specified u acts on the simple module M with type (dim(M)) if dim(M)≤o(u), and as (o(u),1dim(M)-o(u)) otherwise.
Proof.
For n=5, when p is odd, we simply check all simple modules for 2⋅Alt5, and for p=2 we check all simple modules for Alt5 and Sym5.
Next, we deal with n=6. For p=5 we check all simple modules for 6⋅Alt6, and since NG(u)/〈u〉 is cyclic, we only need to know which have dimension at most 6 by Lemma 8, so we get what is above.
For p=3 we check all simple modules for 2⋅Alt6=SL2(9), and the answer will be the same as for SL2(q) in defining characteristic.
For p=2, here we have Alt6, the three extensions of Alt6 by an outer automorphism, so Sym6, PSL2(9) and M10, the central extension 3⋅Alt6, and the last group 3.M10, since the M10 outer automorphism is the only one preserving the centre of 3⋅Alt6. This group is not necessarily well defined, so we give more details now.
Let G be a group of the form 3⋅M10. By a quick computer calculation, G is generated by two conjugates of u for o(u)=4,8 and by three if o(u)=2, by checking this is true for M10.
The only faithful simple modules for G have dimensions 6 and 9, by [16], with the 6 restricting to 3⋅Alt6 as the sum of two non-isomorphic 3-dimensional simple modules. Thus here we are in the situation of Lemma 9, and u2 must act with a single Jordan block of these 3s. This means that o(u2)=4 and so o(u)=8. Furthermore, the only action of u that squares to Jordan block structure (3,3) is (6), so that u does indeed act minimally actively on M of dimension 6.
For the module M of dimension 9, this restricts simply to 3⋅Alt6, and only u of order 8 could act minimally actively on M. This time, we are resigned to constructing the normalizer inside GL9(4) of 3⋅Alt6 and simply computing the action of these elements of order 8, and they do act as (8,1) on the two dual 9-dimensional simple modules.
For G0 a central extension of Alt7, for p=5,7, CG(u)=〈u〉Z(G0) and so we only need dim(M)≤p+1 by Lemma 8. For p=3, we find the two simple modules for 2⋅Alt7 which are minimally active for the conjugacy class of elements of order 3 that have the smallest centralizer. For p=2, the outer automorphism inverts the centre of 3⋅Alt7, so we only need concern ourselves with G one of Alt7, Sym7 and 3⋅Alt7, all of which are easily constructible.
For Alt8, we simply need to consider Alt8 and 2⋅Alt8 for p odd, and Alt8 and Sym8 for p=2, which is easy to do directly.
∎
From now on we let G0 be a central extension of an alternating group Altn for some n≥9. We first need to decide which elements of Symn act minimally actively on the non-trivial composition factor of the permutation module. The next lemma does this.
Lemma 17.
Let G=Symn, and let V denote the non-trivial composition factor of the permutation module for G. If u acts minimally actively on V, then one of the following holds:
u is a single cycle of length pa for some pa≤n, acts on V with type (pa-2) or (o(u),1n-o(u)),
n=2a+2,
a≥2,
p=2,
u has cycle type (2a,2) and acts on V with type (n-2)=(2a),
n=2a+4 with a≥2,
p=2,
u has cycle type (2a,2,2) and acts on V with type (n-2,12),
n=3a+3 with a≥2,
p=3,
u has cycle type (3a,3) and acts on V with type (n-1,1).
Proof.
Let M denote the permutation module for G. Note that V is obtained from M by removing either a single trivial summand if p∤n, or removing a trivial submodule and a trivial quotient if p∣n.
Note that the action of u on M has type the cycle type of u, so if u has more than one cycle of length at least 4, more than two cycles of length at least 3, or more than three non-trivial cycles, then u cannot act minimally actively on V, as V is obtained from M by removing at most two trivials.
Therefore we are left with checking the types (m,1n-m), (m,2,1n-m-2), (m,22,1n-m-4) and (m,3,1n-m-3), with p any prime, 2, 2 and 3, respectively.
First suppose that u fixes a point, so that u∈Symn-1. The restriction of V to Symn-1 is either simple if p∣n, or isomorphic to the permutation module on Symn-1 if p∤n. From this we can use induction to easily see that if u fixes a point and is minimally active, then u acts like a single cycle, and the type of u on V is as above. Thus it remains to check cycles types of the form (m,2), (m,2,2) and (m,3), for m=2a, 2a and 3a, respectively.
If u has cycle type (pa,p) for p=2,3, then up fixes a point, whence its action on V is known from the above working to have p blocks of size pa-1 and p-2 blocks of size 1. Since the action of u on the permutation module has Jordan blocks one of size pa and one of size p, and the action of u on V is a subquotient of this, it must be that u acts with one block of size pa and p-2 of size 1, as needed.
We now need to consider u of type (2a,2,2), which lies inside the subgroup H=Symn-4×Sym4. The permutation module for H is simply the direct sum of the permutation modules for Symn-4 and Sym4, and has as a subquotient of codimension 4 a semisimple module obtained by removing all four trivials. The action of u on the 2-dimensional simple subquotient of the second summand is trivial, since (1,2)(3,4) lies in the kernel of every simple module for Sym4. Thus u acts on this semisimple module with blocks (m-2,1,1), where m=2a, with this semisimple module being itself a subquotient of V. As u2 acts on V with blocks (m2,m2,1,1,1,1), we see that the only possibility for the action of u consistent with both piece of information is that it has type (m,1,1).
∎
Proposition 18.
Let G0 be a central extension of Altn for n≥9, and let u be a p-element of G with G=〈G0,u〉. Let V denote the non-trivial composition factor of the permutation module for G. If u acts minimally actively on a non-trivial simple module M, then (up to outer automorphism in the case n=6) one of the following holds:
G=Altn for p odd and G=Symn for p=2,
u is a single cycle of length pa for some pa≤n and M=V, and u acts with type (n) or (o(u),1n-o(u)),
G=Altn for n=2a+2,
a≥2,
p=2,
u has cycle type (2a,2) and M=V, acting with type (2a),
G=Symn for n=2a+4 with a≥2,
p=2,
u has cycle type (2a,2,2) and M=V, acting with type (n-2,1,1),
G=Altn for n=3a+3 with a≥2,
p=3,
u has cycle type (3a,3) and M=V, acting with type (n-1,1),
G=2⋅Alt9,
p=3,
o(u)=9 and dim(M)=8, acting with type (7,1),
G=2⋅Alt9,
p=7,
o(u)=7 and dim(M)=8, acting with type (7,1).
Proof.
We consider two cases.
Case 1: p odd. We first assume that G0=Altn. We start by checking that there are no simple modules for G0, other than the trivial module and V, that have dimension at most 2n-2. By [14, Theorem 7 and Table 1] for n≥12 we have that this holds, and the dimensions of simple modules are known for n≤11, so we can check that this holds.
We can do the same thing for G0=2⋅Altn: by [20] for n≥12 we have that all faithful representations of G0 are of dimension greater than 2n-2. Thus we need to check 9≤n≤11: for all odd primes the minimal degrees are 8, 16 and 16, with all other faithful modules have dimension larger than 2n-2, unless n=10 and p=5, in which case the minimal degree is 8.
For p=11 this cannot yield a minimally active module as CG(u)=〈u〉⋅Z(G) and so dim(M)≤p+1=12 by Lemma 8.
For p=7, since α(u)=2, we eliminate n=10,11, and for n=9 we see that CG(u)=〈u〉⋅Z(G) and so u acts minimally actively on M if and only if dim(M)≤p+1=8 by Lemma 8 again. Since dim(M)=8, this means they are minimally active.
For p=5, we check that u acts on the 8-dimensional simple modules for 2⋅Alt9 as (4,4), so not minimally active, and for 2⋅Alt10 both classes act as (4,4), so again no minimally active faithful modules, hence none for 2⋅Alt11 either by restriction.
Finally, for p=3 we have o(u)=9. For Alt9 the 8-dimensional module is minimally active, with action (7,1), but for n=10,11 the action on the 16-dimensionals has blocks (9,7), so not minimally active.
Thus if u is a p-element such that α(u)=2, the only simple modules on which u acts minimally actively are the trivial and V, unless n=9 and G0=2⋅Alt9. Note that if o(u)=3, then since α(G)≤n2 by [10, Lemma 6.1], we have that dim(M)≤3n2 by Lemma 5
(iv), so M is either trivial or V; hence we will assume that o(u)≥5.
Let G0=Altn or G0=2⋅Altn for some n≥9. By Lemma 11, if the image of u in Altn has no fixed points, then α(u)=2 and we are done by the previous paragraph, so we may assume that the image of u in Altn lies in Altn-1; we restrict a simple module M on which u acts minimally actively to H=Altn-1 or H=2⋅Altn-1. Suppose that H=Altn-1 first. If a trivial submodule or quotient lies in this restriction, then M is a composition factor of the permutation module on the cosets of H, so is either trivial or V. Moreover, since the composition factors of the restriction are minimally active, we know that the composition factors of M↓H are either trivial or copies of VH, the corresponding simple module for H. Since u acts on VH with a block of size at least o(u)-2, if there were more than one composition factor of M↓H isomorphic with VH, then these two large Jordan blocks cannot form blocks of the form o(u),1a unless 2(o(u)-2)≤o(u)+1, i.e., o(u)=5, and then dim(VH) would need a block of size 3, only possible if H=Alt5, but then n=6, which is not allowed. Thus M↓H has at most one copy of VH, and can have no trivials as they would have to be submodules or quotients, so dim(M)≤n-1, as needed.
If H=2⋅Altn-1, then all composition factors of the restriction of M to H are non-trivial. If n=9, then from Proposition 16 we see that p=7. Since Alt9 is generated by two 7-cycles, M is therefore a module that appears in the computations above. If n=10, then again p=7, this time by induction and using this proposition. Again, α(u)=2, and so M does not exist as it does not appear above. Finally, for n≥11 there can be no examples as there are no examples for n=10. This completes the proof.
Case 2: 𝐩=2. Here we do not need to consider 2⋅Altn but do need to consider G=Altn and G=Symn. If α(u)=2, then again we need that dim(M)≤2n-2, in fact merely dim(M)≤2(o(u)-1). From [14, Theorem 7 and Table 1] we have dim(M)>2n-2 for M≠k,V for all n≥15, and for n≤14 we have that o(u)≤8, so we just need dim(M)≤14, for which there are two modules for G=Alt9 with this property, both of dimension 8 (but not isomorphic to V). However, Alt9 does not contain elements of order 8, so these cannot be examples.
If α(u)=2, so that M=V, then by Lemma 17 we know that u acts minimally actively on V if and only if we are in cases (i)–(iii) of the proposition. Thus we may assume that α(u)>2, and in particular we cannot be in the situations given in Lemmas 11 and 12.
Suppose that u has at least three 2-cycles and does not have order 2: we can write u=u1u2, where the supports of the ui are disjoint, and where u1 has cycle type (2a,2,2,2) for some a≥2. Write Hi for the symmetric group on the support of ui, H=H1H2=H1×H2, and note that the restriction M↓H is minimally active. The simple kH-modules are tensor products of simple kHi-modules, and by Lemma 5
(v) a tensor product of two non-trivial simple modules is not minimally active (unless they both have dimension 2), we see that the composition factors of M↓H are (minimally active) simple modules for one of the Hi. But H1 has no non-trivial minimally active modules, so H1 lies in the kernel of M, clearly nonsense as M is faithful.
Suppose that u has at least one 2-cycle and at least two cycles of length at least 4. Again, write u=u1u2, this time with u the product of all of the cycles of length at least 4, unless there are exactly two of length exactly 4, in which case add another 2-cycle. (If u has cycle type (4,4,2), then α(u)=2 by Lemma 12.) Defining the Hi and H as above, we again note that no non-trivial minimally active modules exist for H1, and so get the same contradiction. Since every fixed-point-free element has one of these properties, we have covered all fixed-point-free cases.
Thus u fixes a point and lies inside H=Symn-1. As for p odd, we restrict to H and note that the exact same proof works, as long as n≥10. In order to apply the argument for the case p odd to p=2, we need to exclude the case G0=Alt9, as for Alt8 there are minimally active simple modules other than k and V. However, we already checked Alt9, so we may do this. Thus M↓H has at most one copy of V and possibly trivial factors, and therefore M is a submodule of the permutation module on H, i.e., M=k or M=V, as needed.
∎
5 Lie type in defining characteristic
In the notation of the end of Section 2, this section considers G0 a central extension of a simple group of Lie type in characteristic p.
Let G be a simple, simply connected algebraic group defined over the field k, and let F be a Frobenius morphism on G. With the exceptions of a few quasisimple groups (e.g., 2⋅Ω8+(2), 3⋅PSL2(9), and so on) if G0 is a quasisimple group of Lie type, then for some choice of G and F above, G0=𝑮F. Moreover, from [7, Table 6.1.3] we see that if Z(G0) is a p′-group, which we require in order to have a faithful irreducible module, then G0 is always 𝑮F (unless G0 is an extension of Sp4(2)′=Alt6, which was examined in the previous section). Hence for this section we can always take G0 to be the fixed points of G under F.
Furthermore, every simple kG0-module is the restriction of a simple k𝑮-module, and by Steinberg’s tensor product theorem, every simple module for G0 is a tensor product of Frobenius twists of p-restricted simple modules. Since tensor products of simple modules cannot be minimally active unless they have dimension 1 or 2 by Lemma 5
(v), we thus consider p-restricted simple modules for G.
As every unipotent class of G appears in G0, when checking the action of a unipotent element on a given p-restricted simple module, we can use either G or any of the quasisimple groups G0 over the various ground fields. For explicit calculations, we of course will usually choose G0 to be the smallest group, so over the field 𝔽p.
If u∈G0, then the problem has been almost completely solved already, in [29] and [30], which dealt with all types apart from C and D in characteristic 2. Here we will finish that last case, and also consider the case where u induces an outer automorphism on G0. The next proposition completes the proof for u∈G0. (In this proposition, (iv) and (v) can be viewed as subcases of (ii), using the isomorphisms B2≅C2 and A3≅D3, but we separate them out for clarity.) We use the notation L(λ) for highest weight modules, using the same conventions as in for example [29] and [30] above. For weights of small rank we can write the weight in full, e.g., 110, but for larger rank it is more clear to write as a sum, so the previous example would be λ1+λ2.
Proposition 19.
Let G0 be a quasisimple group of Lie type, and let u∈G0 be a non-trivial unipotent element. If M is a simple kG0-module on which u acts minimally actively, then up to outer automorphism of G0, one of the following holds:
G0=SL2(pa),
M=L(i) for 0≤i≤p-1, or M=L(1+pj) of dimension
4
for some 1≤j≤a2 and p odd, u has type (i+1) and (3,1), respectively,
G0 is of type A,
A2,
B or C for all primes, D,
D2 or D43 for p odd, B22 for p=2,
M is the natural module,
p>3,
G0=SL3(pa) or SU3(pa),
u has type (3) on the natural module, M=L(2λ1) of dimension
6,
u has type (5,1),
p is odd, G0=SL4(pa) or SU4(pa),
u has type (4) or (2,2) on the natural module, M=L(λ2) of dimension
6,
u has type (5,1) or (3,13), respectively,
p is odd, G0=Sp4(pa),
u has type (4) or (2,2) on the natural module, M=L(λ2) of dimension
5,
u has type (5) or (3,12),
p is odd, G0=Spin7(pa),
u is regular unipotent, M=L(λ3) of dimension
8,
u has type (7,1),
G0=G2(pa) or G22(32a+1),
u is regular unipotent, M=L(λ1) of dimension
7
(or
6
for p=2), u has type (7) (or (6) for p=2).
Proof.
If G0=SL2(pa), then for M=L(λ)p-restricted we see that u always acts with a single block, and for L(λ) not p-restricted we use Lemma 5
(v), which shows that p is odd and M is the tensor product of two 2-dimensional modules, i.e., L(1+pj). By applying a field automorphism we may assume that 1≤j≤a2.
For G0 of type A, or types B, C and D and p odd, [29, Theorem 1.3] gives (ii)–(vi). (Note that the case of p=3 for G0=SL3(pa) was erroneously included in [29], but can be excluded by Lemma 5
(vii).) For G0 of exceptional type [30] shows that only (vii) occurs, so we are left with types B/C and D in characteristic 2.
Let G0=Sp2n(2a) first. By [23, Chapter 4] every Jordan block of odd size appears an even number of times, so there can be no element of order 4 in G0 that powers to a transvection. Thus if u is not itself a transvection, no power of u is one. By [10], α(u)≤n+3 if u is not a transvection, and of course since G0 acts on a 2n-dimensional space the order of u is at most 2an, where an is the smallest power of 2 that is at least n. By Lemma 5
(iv) we have
dim(M)≤α(u)⋅(o(u)-1)≤(n+3)(2an-1).
(If u is a transvection, then α(u)=2n+1, and so dim(M)≤2n+1, so that M is the natural module.) By [24, Theorems 4.4 and 5.1], if n≥8, then M is one of the standard module, its symmetric square (does not occur for p=2), or its exterior square with a trivial removed (with two trivials removed if n is even).
For n≥3, by Lemma 6 we see that this exterior square cannot be minimally active for u, and for n=2 the simple modules for Sp4(2) are the trivial, the natural and its image under the graph automorphism, and the Steinberg. Since Sp4(2)=Sym6, we use Proposition 16. Therefore we can assume that M is neither the natural L(λ1) nor the non-trivial factor of its exterior square L(λ2), and that n≥3.
For 3≤n≤7, we use [24, Theorem 4.4] to get the following table of low-dimensional modules, with modules listed in order of increasing dimension.
| n | Bound | Modules |
| 3 | 42 | 100, 001, 010 |
| 4 | 49 | 1000, 0001, 0100, 0010 |
| 5 | 120 | 10000, 00001, 01000, 00100 |
| 6 | 135 | λ1, λ2, λ6 |
| 7 | 150 | λ1, λ2, λ7 |
As we have excluded L(λ1) and L(λ2), we need to consider the spin module L(λn) for Sp2n(2) for 3≤n≤7, and the modules L(0010) and L(00100) for Sp8(2) and Sp10(2), respectively.
For Sp8(2), L(0010) is 48-dimensional, and since α(u)≤7 for u not a transvection, we must have that o(u)=8 if u acts minimally actively. If u has type (6,12), (6,2) or (8) on the natural module, then an easy computer calculation shows that u has type (84,62,22), (84,62,22) or (86), respectively, on L(0010), so cannot be minimally active.
For Sp10(2), L(00100) has dimension 100, and since α(u)≤8 for u not a transvection, we must have that o(u)=16 if u acts minimally actively, so u is the regular unipotent element. This element has type (164,14,10,62) on L(00100), and so is not minimally active.
Finally, let M=L(λn). Note that the restriction of M to the Sp2n-2-parabolic has two composition factors, both isomorphic to L(λn-1). For n=3, we compute the Jordan block structure of all unipotent elements on L(λ3) and get the following table.
| Class | Action on L(λ1) | Action on L(λ3) |
| C1 | 2,14 | 24 |
| A1 | 22,12 | 22,14 |
| A1(2) | 22,12 | 24 |
| A1+C1 | 23 | 24 |
| C2 | 4,12 | 42 |
| A2 | 32 | 32,12 |
| C3(a1) | 4,2 | 42 |
| C3 | 6 | 6,2 |
In particular, we see that there are no minimally active elements in Sp6(2) on L(001). Thus we proceed by induction on n. If u in Sp2n(2) acts minimally actively on L(λn), then place u inside an Sp2n-2(2)-parabolic: it must act minimally actively on the indecomposable module for this group with socle L(λn-1), whence in particular it acts minimally actively on this submodule by Lemma 5
(i). Since there are no non-trivial elements of Sp2n-2(2) that act minimally actively on L(λn-1) by induction, u must act trivially on L(λn-1), whence o(u)=2. One can see this either because the unipotent radical of the parabolic is elementary abelian, or because the action of u on L(λn) must have blocks only of size 1 and 2, since there are two composition factors of the restriction of L(λn) to the parabolic. At any rate, this is impossible since α(u)≤2n+1 and
dim(M)=2n>2n+1≥α(u)⋅(o(u)-1)
by Lemma 5
(iv).
We therefore need to consider groups of type D now. Let G0=Ω2n±(2), and note that α(u)≤n+3 by [10, Theorem 4.4] (as transvections induce the graph automorphism on G0, so do not lie in G0 itself). By placing G0 inside Sp2n(2), we see that no unipotent element can act minimally actively on L(λ2) since this is the restriction of the corresponding module for Sp2n(2). (See for example [26, Table 1, MR4].)
Using the bound dim(M)≤α(u)⋅(o(u)-1) from Lemma 5
(iv), and [24, Theorems 4.4 and 5.1], we get the possible minimally active modules are the natural L(λ1), and L(λ2) (already eliminated above) for n≥9, and for 4≤n≤8 we get the table below.
| n | Bound | Modules |
| 4 | 49 | (1000, 0010, 0001), 0100, (0011, 1010, 1001) |
| 5 | 56 | λ1, λ2, (λ4, λ5) |
| 6 | 135 | λ1, λ2, (λ5, λ6) |
| 7 | 150 | λ1, λ2, (λ6, λ7) |
| 8 | 165 | λ1, λ2, (λ7, λ8) |
The brackets indicate the groupings under the outer automorphism group. For n≥5 we need to check the two half-spin modules, but for n=4 there are other modules to check, with the 48-dimensional modules L(0011) and so on only occurring because our bound for α(u) is lax: checking by computer that α(u)=2 when o(u)=8 inside Ω8+(2), we can therefore exclude them.
For n=5, we note that D5 lies inside E6 acting as (up to automorphism) L(0)⊕L(λ1)⊕L(λ5) on the minimal module for E6. As the dimension of L(λ5) is 16, we therefore need at least 17-o(u) trivial Jordan blocks in the action of the corresponding unipotent class of E6 on the minimal module: examining [22, Table 5], only the class A1 of elements of order 2 have enough blocks of size 1, and of course if o(u)=2, then dim(M)≤2n<16 (using [10, Theorem 4.4] and Lemma 5
iv) which does not work either. For n≥6, we use induction, exactly the same as for type C. This completes the proof.
∎
We now consider the case where u induces an outer automorphism on G0. We start with u inducing a graph automorphism on an untwisted group G0. In this case, p=2 unless G0 is type D4, and then p=2,3. We will then examine field and mixed field-graph automorphisms on untwisted groups, and finally how the automorphisms of the twisted groups compare with those of the untwisted groups.
In this proposition, (iii) may be thought of as the case n=3 of (v), but we separate them for clarity.
Proposition 20.
Let G0 be a quasisimple group of Lie type in characteristic p, and suppose that u lies in the coset of a graph automorphism on G0. If M is a minimally active, non-trivial simple module for G, then one of the following holds:
G=SLn(2a).2,
M=L(λ1)⊕L(λn-1) has dimension 2n, and u2 is the regular unipotent element,
G=SL3(2a).2,
M=L(11) has dimension
8
, and u has order
8,
G=SL4(2a).2,
M=L(010) has dimension
6
, and there are four possible classes for u,
G=Sp4(2a).2 for a odd, M=L(10)⊕L(01) has dimension
8
, and u2 is the regular unipotent element,
G=SO2n+(2a) for n≥4 and M is the natural module L(λ1).
Proof.
Since u induces a graph automorphism on G0, we have that G0 is untwisted by [7, Theorem 2.5.12 (f)].
Suppose that u induces a graph automorphism on G0 and that p=2, so that G/G0 has order 2. We go through each possibility in turn, of type A, type D, C2, F4 and E6. (Note that G2 only possesses the graph automorphism of order 2 when p=3, so this case does not occur.) Notice that, if M↓G0 is simple, then by Lemma 10 we may assume that it is 2-restricted or G0=SL3(2a) and M↓G0 is the product of two 3-dimensional simple modules: but this is never graph stable, so we can ignore this case.
If G0=E6(2a), then, since o(u)≤32 and α(u)≤9, we have the bound dim(M)≤9⋅31=279 if M is minimally active, by Lemma 5
(iv). The adjoint module L(λ2) is the only (non-trivial) graph-stable simple module with dimension at most 279, and the action of unipotent elements on this is given in [22]. For u to be minimally active, u2∈G0 must have at most two non-trivial Jordan blocks, but this is not the case. If M↓G0 is not simple, then by Lemma 9u2 acts with a single Jordan block, which is not possible by Proposition 19.
For G0=F4(2a), the exponent of the Sylow 2-subgroup of G0 is 16, thus o(u)≤32. Since α(u)≤8 from [10, Theorem 5.1], this gives dim(M)≤248. The dimensions of the simple modules are 1,26,26,246,246, and 676 and above. The (non-trivial) modules of dimension at most 246 are not graph-stable, and so as in the previous case we apply Lemma 9 and Proposition 19 to prove that no examples occur.
For G0=Sp4(2a) (we may assume that a≥2 since a=1 yields Sym6, which has been considered already), there are only four 2-restricted modules: the trivial, the two 4-dimensional modules L(10) and L(01), swapped by the graph automorphism, and L(11): the regular element acts with four blocks of size 4 on the L(11) so this cannot extend to a minimally active module for G, and if u squares to the regular then L(10)⊕L(01) is minimally active by Lemma 9, since u2 acts with a single Jordan block on L(10). However, u can only induce a graph automorphism of order 2 if the Sylow 2-subgroup of Out(G0) has order 2 by [7, Theorem 2.5.12 (e)], with the graph automorphism squaring to a field automorphism in the other cases.
For G0 of type D, note that G0.2=SO2n+(2a) lies inside Sp2n(2a), with dim(M)≤(n+3)⋅(o(u)-1) unless u is a transvection by [10, Theorem 4.4], and so we get the same bound as for the unipotent elements for the simple group of type C in the proof of Proposition 19. Using the tables from [24], we see that for n≥5, every simple module for G that satisfies the bound on dim(M) is the restriction of a simple module for type C, and hence is minimally active only if the module is for type C. This yields only the natural module, which is of course minimally active. When n=4, we get for G0 the modules 0000 (trivial), 1000 (natural), 0010⊕0001 (sum of the two half-spins), 0100 (exterior square of natural), and 0011 of dimension 48. These are all also the restriction of a module for Sp8(2a), and so we are again done.
The last case for p=2 is G0=SLn(2a). If M↓G0 is not simple, then by Lemma 9 we have that v=u2 acts as a single Jordan block on each factor: thus v is the regular element and M is the sum of the natural and its dual.
Thus we may assume that M↓G0 is simple, i.e., M↓G0 is a graph-stable simple module, and 2-restricted by our discussion at the start of the proof. If v has maximal order in G0, then α(u)≤4 by Lemma 14, and writing an for the smallest power of 2 that is at least n, we have that dim(M)≤4⋅(2an-1). We now use [24, Theorems 4.4 and 5.1] to get that L(λ1+λn-1) is the only graph-stable module of dimension at most this for n≥7. For SL6(2a) we have the module L(λ3), with L(0110) for SL5(2a). For n≤4 there are several possibilities and we deal with them later.
If u does not have maximal order, then o(u)≤an and α(u)≤n by [10, Theorem 4.1], so that dim(M)≤n(an-1)<2n2. We again use [24] to see which graph-stable modules we need to consider: doing so yields smaller bounds than the previous case, and so we need only consider n=3,4, and the specific modules for larger groups above.
We quickly show that L(λ1+λn-1) is not minimally active if n≥5: it is obtained from the tensor product of the natural and its dual by removing at most two trivial factors, and so if u inducing a graph on L(λ1+λn-1) is minimally active, then v acts with at most two non-trivial blocks on the module. As in the proof of Lemma 6, if v acts with at least two non-trivial blocks on L(λ1), then the action of v on the tensor square has at least eight non-trivial blocks, whence u cannot act minimally actively if one removes at most two trivials. If v acts with a trivial block and a block of size at least 3, then v acts on the tensor square with blocks at least of size (4,4,3,3,1,1), so u cannot be minimally active when removing two trivials. Similarly, if v contains a block of size at least 4, then v acts on the tensor square with at least four blocks of size at least 4, and if v acts on the natural with a block of size 2 and at least three trivial blocks, then v acts on the tensor square with at least five blocks of size at least 2, hence again u cannot be minimally active.
The actions of the classes of SL6(2) on L(λ3) are given in the following table.
None of these can be the square of a minimally active element, so we are done.
| Type on L(λ1) | Type on L(λ3) |
| 2,14 | 26,18 |
| 22,12 | 28,14 |
| 23 | 210 |
| 3,13 | 36,12 |
| 3,2,1 | 42,32,22,12 |
| 32 | 44,14 |
| 4,12 | 44,22 |
| 4,2 | 44,22 |
| 5,1 | 72,32 |
| 6 | 82,22 |
For SL5(2), the dimension of L(0110) is 74, so if o(u2)≤4, then this module fails the bound α(u)⋅(o(u)-1), as α(u)≤5. However, the only element v of SL5(2) with order 8 is the regular unipotent element, and for this one α(v)=2 by Lemma 13, so this module cannot be minimally active.
We are thus left with n=3,4. For n=4, we examine Proposition 16, which states that the 6-dimensional module L(λ2) is the only graph-stable minimally active module, and here there are several classes that work. For n=3, we simply check the simple modules for PGL2(7)=SL3(2).2, and an element of order 8 acts with a single Jordan block on both L(10)⊕L(01), in line with the proposition, and also on the 8-dimensional module L(11), as seen in [5, Theorem 1.2].
For p=3, we only have the group Ω8+(3a) to consider, and since the graph automorphism of order 3 permutes the central involutions of the Spin group regularly, we may assume that G0 is simple. If M is minimally active and M↓G0 is not simple, then u3 must act with a single Jordan block on each composition factor of M↓G0 by Lemma 9, but this does not occur by Proposition 19. Thus M↓G0 is simple: the smallest non-trivial, graph-stable simple module has dimension 28, and the next smallest has dimension 195 (see [24, Appendix A.41]), and since α(u)=2 if o(u)=9 and α(u)≤4 if o(u)=3 for G0=PΩ8+(3), we have that dim(M)=28 and M=L(λ2). Since u is minimally active, u3 has at most three non-trivial Jordan blocks. However, of the 27 non-trivial unipotent classes of 3-elements in Ω8+(3), all have at least four non-trivial blocks on L(λ2). Thus there is no candidate for M.
∎
Having dealt with the case where u induces a graph automorphism, we are left with the case where u induces either a field automorphism or a mixed field-graph automorphism on the untwisted group G0.
Proposition 21.
Let G0 be a quasisimple group of Lie type in characteristic p, and suppose that u induces an outer automorphism on G0 that is not a graph automorphism. If M is a minimally active simple module for G, then up to outer automorphism of G one of the following holds:
p=2,
G=SL2(22a).2,
M=L(1+2a) of dimension
4,
p=3,
G=SL2(33a).3,
M=L(1+3a+32a) of dimension
8,
p=2,
G=SL3(22a).2,
u induces either a field or the product of a field and graph automorphism on G0,
M=L(1+2a,0) or M=(1,2a), respectively, both of dimension
9,
p=2,
G=SU3(22a).2,
u induces the unique outer automorphism of order
2
on G0,
dim(M)=9 is such that M↓G0 is simple,
G=G0.t,
M↓G0=M1⊕⋯⊕Mt with ut∈G0 acting on each Mi with a single Jordan block, the possibilities for which are given in Proposition 19.
Proof.
We begin with the case where G0 is untwisted.
Suppose that M is minimally active and that M↓G0 is the simple module L(λ). By Lemma 10, either L(λ) is (up to Frobenius twist) a p-restricted module, or p=2 and G0=SL3(2a), or p=2,3 and G0=SL2(pa), with M a product of p twists of the natural module. If u induces a field automorphism, then u replaces a highest weight λ=a1λ1+⋯+anλn with pαa1λ1+⋯+pαanλn for some α≥1, and since graph automorphisms permute the ai, for u to fix L(λ) it cannot be p-restricted.
In the remaining cases of SL3(2a) and SL2(pa) for p=2,3, we have from Lemma 10 that u has order p and M is the tensor product of the modules in a single orbit under the action of u: for G0=SL2(pa), this orbit is clear, whereas for SL3(2a) if u is a pure field automorphism, we get that L(1,0) and L(2a,0) form an orbit, and if u is the product of the field and the graph, it is L(1,0) and L(0,2a), yielding the modules in the statement of the proposition.
We therefore have that M↓G0 is not simple, and is the sum of t simple factors M1,…,Mt, each stabilized by ut. By Lemma 9, ut acts on each Mi with a single Jordan block, and since ut cannot act as a pure graph automorphism on the Mi, we must therefore have that ut∈G0, and so the conclusion of the proposition holds here as well.
Now suppose that G0 is twisted, so that u must be a field automorphism: if M↓G0 is simple, then it cannot be p-restricted, so as with the untwisted case we get that G0 is of type A2 and we proceed similarly to the case of SL3(2a). If M↓G0 is not simple, then we get the same proof as for the untwisted case, yielding the result above.
∎
There are many examples of (v) above, and some that may appear to be but are not. For example the group D43(3).3 with M of dimension 24, is the sum of three 8-dimensional modules permuted by the graph automorphism. Each 8-dimensional module for D43(3) possesses minimally active elements by Proposition 19, but since the simple group of type D does not contain an element acting with a single block of size 8, there can be no minimally active u for M.
As another negative example, we consider G=G22(3), which has a module M of dimension 27 where the derived subgroup G′ acts on M as the sum of three blocks, and G′ contains an element v (of order 9) acting on each factor with a single Jordan block, but there is no element u∈G such that u3=v (as then the Sylow 3-subgroup would be cyclic).
We should enumerate which possibilities from Proposition 19 actually have an element acting with a single block, so that they can be inputs into case (v) in Proposition 21. For the group G0=SL2(pa), we have that u acts as a single block on L(i) for i<p, but acts as (3,1) on L(1+pi). For G0 of type A, A2, B, B2, and C, there is an element acting with a single Jordan block, but not for those of type D. We can include the 5-dimensional module for Sp4(pa) in those above via the isomorphism B2≅C2, and we also have G2 with the minimal module.
Hence, if G0=SUn(pp) for some n,p for example, one may form the group G=SUn(pp).p, and let u be an element of g whose pth power is the regular unipotent element of G0. In this case the hypotheses of (v) are satisfied, so this is an example.
7 Groups of Lie type in cross characteristic: preliminaries
In this section we tackle groups of Lie type in characteristic r where r≠p, so let q be a power of r. We specifically exclude the case where the group is also a group in characteristic p, so for SL3(2) for example, p≠2,7. If G0 is a classical group and M is a Weil representation, or if G0=PSL2(q), with u inducing an inner-diagonal automorphism on G0 in both cases, then all minimally active u are classified in [5], but the case where u induces particularly a non-diagonal outer automorphism on M is missing. In this section we give some notation and introduce some previous results, particularly on minimal dimensions of irreducible representations.
Let G0 be a quasisimple group of Lie type, defined over the field 𝔽q, let r be the prime dividing q, and let p≠r be another prime. With a few exceptions given in [7, Table 6.1.3], G0 is (a quotient of) the fixed points of a Frobenius endomorphism of a simple, simply connected algebraic group in characteristic r.
Note that those groups of Lie type that are isomorphic to alternating groups have already been covered in Proposition 16, and we will ignore them from now on. We will also ignore cases where G0 is isomorphic to a group in characteristic p, so under the isomorphisms:
We find all simple, minimally active modules for classical G with G0 possessing one of the central extensions given in [7, Table 6.1.3] now. For G0 of exceptional type, see Proposition 36 except for E62(2), for which standard arguments work.
In the next result, if there is more than one central extension n⋅G (for example, G=PSL3(4) has more than one 4-fold extension), we use the numbering of these from the Atlas [2], and we use [16] for information about the existence of certain simple modules for central extensions.
Proposition 24.
Let G0 be a central extension of one of the following groups: PSL3(2), PSL3(4), PSU4(2), PSU4(3), PSU6(2), Sp6(2), Ω7(3), Ω8+(2). Let u be a p-element of G such that G=〈G0,u〉. If u acts minimally actively on a non-trivial simple module M, then one of the following holds:
G=SL3(2),
p=3,
o(u)=3 and dim(M)=3 (two representations) requires 𝔽9⊆k,
G=2⋅SL3(2),
p=3,
o(u)=3 and dim(M)=4 (two representations) requires 𝔽9⊆k,
G=6⋅PSL3(4),
p=5,7,
o(u)=p and dim(M)=6 (two representations), with u acting as (5,1) or (6),
G=41⋅PSL3(4),
p=7,
o(u)=7 and dim(M)=8 (four representations), with u acting as (7,1),
G=PSU4(2),
p=5,
o(u)=5 and dim(M)=5 (two representations) requires 𝔽25⊆k or dim(M)=6 (one representation), with u acting as (5) or (5,1),
G=2⋅PSU4(2),
p=5,
o(u)=5 and dim(M)=4 (two representations) requires 𝔽25⊆k, with u acting as (4),
G=61⋅PSU4(3),
p=5,7,
o(u)=p and dim(M)=6 (two representations), with u acting as (5,1) or (6),
G=31⋅PSU4(3),
p=2,
o(u)=8 and dim(M)=6 (two representations), with u acting as (6),
G=31⋅PSU4(3).22,
p=2,
u can have order
2,
4,
8
(but not all elements of orders
2
or
4
), and dim(M)=6 (two representations), with u acting as (2,14),
(4,12) and (6),
G=Sp6(2),
p=3,
o(u)=3,9 and dim(M)=7, with u acting as (3,14) or (7) (minimally active elements of order
3
are from the smallest conjugacy class),
G=Sp6(2),
p=5,7,
o(u)=p and dim(M)=7, with u acting as (5,12) and (7),
G=2⋅Sp6(2),
p=3,7,
o(u)=7,9 and dim(M)=8, with u acting as (7,1),
G=2⋅Ω8+(2),
p=3,
u has order
9
, or u has order
3
with centralizer Z6×PSp4(3), and dim(M)=8,
G=2⋅Ω8+(2),
p=5,
u has order
5
with centralizer Z10×PSL2(5), and dim(M)=8,
G=2⋅Ω8+(2),
p=7,
o(u)=7 and dim(M)=8.
Unless specified, the field of definition of these representations is the smallest k such that |Z(G0)| divides |k×|.
Proof.
If G0 has PSL3(2) as a quotient, then p≠2,7, as these were considered in Section 5. Thus p=3, G=SL2(7), o(u)=3, and CG(u)=〈u〉⋅Z(G), so that M is minimally active if and only if dim(M)≤p+1=4 by Lemma 8.
Now let G0 be a central extension of PSL3(4), and to begin let p=3. The Sylow 3-subgroup of PGL3(4) has exponent 3: PSL3(4) has a unique class of elements u of order 3 and is generated by two conjugates of them, so if u acts minimally actively, then dim(M)≤4, but the minimal dimension for a simple module for (4×4)⋅PSL3(4) is 6. (The Schur multiplier of PSL3(4) is 4×4×3.)
Alternatively, u could lie outside PSL3(4) in G, and then G is generated by three conjugates of u, so dim(M)≤6. However, we cannot form a central extension of PGL3(4) by a 2-group as the outer automorphism acts transitively on the involutions in the Z4×Z4 Sylow 2-subgroup of the Schur multiplier, and the minimal dimension for PGL3(4) is 19.
When p=5, the normalizer of a Sylow 5-subgroup of PSL3(4) is D10, so we apply Lemma 8 to see that M is minimally active if and only if dim(M)≤6. Similarly, the normalizer of a Sylow 7-subgroup of PSL3(4) is Z7⋊Z3, so again dim(M)≤8 by Lemma 8. There are 6-dimensional representations of 6⋅PSL3(4) modulo 5 and 7, and 8-dimensional representations of 41⋅PSL3(4).
If G0 is a central extension of PSU4(2)=PSp4(3), then p=5.
For p=5, the Sylow 5-subgroup has order 5, generated by u, and CG(u)=〈u〉⋅Z(G), so that M is minimally active if and only if dim(M)≤p+1=6. There are modules for PSU4(2) of dimensions 5 and 6, and of 2⋅PSU4(2) of dimension 4, completing the proof.
Let G0 be a central extension of PSU4(3), so that p=2,5,7. For p=5,7, the Sylow p-subgroup is of order p, generated by u, and CG(u)=〈u〉⋅Z(G), so that dim(M)≤p+1 by Lemma 8. There is a module of dimension 6 for 61⋅PSU4(3), so this is minimally active for both primes. For p=2, as Out(G0) is D8 and the Schur multiplier is 3×3×4 there are many potential groups G.
If G0=PSU4(3), then from [16] we see that dim(M)=20 or dim(M)≥34. The order of u∈G is 2,4,8 from [2], and for u∈G0, α(u)=2 if o(u)=4,8, with α(u)=3 if o(u)=2. As Out(G0)=D8, u4∈G0, so the only way that u can act minimally actively is if o(u)=8, u2∉G0, and α(u)=3. However, by constructing Aut(G0) in Magma, we check that α(u)=2 for all u of order 8, hence there are no non-trivial minimally active modules if Z(G0)=1. In Aut(G0), we have α(u)=2 if o(u)=8, α(u)≤4 if o(u)=4, and α(u)≤6 if o(u)=2.
Thus G0 is either 31⋅PSU4(3) or 32⋅PSU4(3). In the second case, we have dim(M)≥36, so we again see that there are no minimally active modules by the above computations for α(u), as dim(M)≤14 for M to be minimally active. Thus G0=31⋅PSU4(3), and from [2, p. 53] we see that the only outer automorphism that centralizes G0 is 22, so we let G be either G0 or G0.22.
The only non-trivial simple module for G0 of dimension at most 14 has dimension 6 (two up to duality), and this extends to G0.22. Inside G0 elements of order 8 act with type (6), and in G0.22 (modulo a central involution, this is the complex reflection group G34 in Shephard–Todd notation) there are elements of orders 2, 4 and 8 (with the last one not in G0) that act with types (2,14), (4,12) and (6), respectively.
Let G0 be a central extension of PSU6(2), so that p=3,5,7,11. If p=7,11, then CG0(u)=〈u〉⋅Z(G0), so that dim(M)≤p+1 if and only if M is minimally active by Lemma 8. However, dim(M)≥21 for all odd p by [21], a contradiction. If p=5, then o(u)=5 and α(u)=2, so that dim(M)≤8 for minimally active M, another contradiction. If p=3, then there is an outer automorphism of order p, but in PGU6(2) the exponent of the Sylow 3-subgroup is still 9, so if α(u)=2 when o(u)=9 and α(u)≤10 for o(u)=3, then we are done. The former follows by a computer calculation, and the latter follows from [10, Theorem 4.1].
Let G0 be a central extension of Sp6(2), so that p=3,5,7. The Sylow 7-subgroup of G0 is cyclic, and if u is a generator for it, then CG0(u)=〈u〉⋅Z(G), so that dim(M)≤p+1=8 if and only if M is minimally active. There is a 7-dimensional simple module for Sp6(2) and an 8-dimensional simple module for 2⋅Sp6(2), so these are minimally active. If p=5, then the Sylow 5-subgroup P has order 5, and α(u)=2 for u of order 5, yielding dim(M)≤8 for minimally active modules. The module of dimension 7 is minimally active, with u of type (5,12), but on the module of dimension 8 the action has type (42).
For p=3, if o(u)=9, then α(u)=2, and if o(u)=3, then α(u)≤4, so dim(M)≤16 for M minimally active. There are three such non-trivial simple modules, of dimensions 7, 8 and 14. The 14-dimensional module is not minimally active, but elements of order 9 act with a single Jordan block on the other two modules. In addition, elements of order 3 from the smallest class, with centralizer of order 2160 in Sp6(2), act on M of dimension 7 with type (3,14).
The next group is G0 a central extension of Ω7(3), with primes 2,5,7,13. From [21], dim(M)≥27 if M is non-trivial. Note that o(u)≤13, and α(u)=2 if o(u)≥5, α(u)≤3 for o(u)=4, α(u)≤4 if o(u)=3 and α(u)≤7 if o(u)=2. Thus there are no minimally active modules for G0. However, Out(G0) has order 2, and from [2] we see that the exponent of the Sylow 2-subgroup of Aut(Ω7(3)) has order 8; in this case, α(u) is as before if o(u)=2,8, and α(u)≤4 if o(u)=4. Again, there can be no non-trivial simple minimally active modules.
The final group on our list is Ω8+(2), where p=3,5,7. For G=2⋅Ω8+(2), α(u)=2 for o(u)=5,7,9, and α(u)≤4 for o(u)=3. The 8-dimensional simple module for G is minimally active for p=7, with type (7,1), and for p=5 the three classes of elements of order 5 have type (5,1,1,1), and (4,4) twice, with the two classes of elements of order 5 having centralizer Z5×SL2(5) and the minimally active one having centralizer Z10×PSL2(5).
For p=3, we need to consider G=2⋅Ω8+(2) and Ω8+(2).3: the former case is easy, with elements of order 9 acting on the 8-dimensional module as (7,1), and one of the three classes of elements of order 3 with centralizer of order 155,520 have type (3,15) and the other two having type (24); again these two have centralizer Z3×Sp4(3), and the one we want has centralizer Z6×PSp4(3). (There are two classes of elements of order 3 with smaller centralizer.)
To deal with Ω8+(2).3, note that from [16] we get that dim(M)≥28, and o(u)=3,9. Since α(u)≤4 by [10, Theorem 4.4], elements of order 3 cannot work, and elements u of order 9 must cube to an element v=u3 of order 3 in G0 that acts on M with at most three blocks of size 3. Since dim(M)=28 or dim(M)≥48, we have dim(M)=28, and the five conjugacy classes of elements of order 3 act on M as (36,110),(37,22,13) and (39,1), so it cannot be minimally active. This completes the proof for Ω8+(2).
∎
Because of Proposition 24, if G is classical, then we may take G0 to be a quotient of one of the groups SLn(q), SUn(q), Sp2n(q), Spin2n+1(q) and Spin2n±(q). The order of G0(q) is given by a polynomial
qN∏iΦi(q)ai,
where N and the ai are integers, and Φi denotes the ith cyclotomic polynomial. If u is a p-element and p∤q, then p divides one of the Φi(q); let d denote the order of q modulo p, so that p∣Φd(q) and p∤Φe(q) for all 1≤e<d. (If p=2, we let d be the order of q modulo 4.) This next well-known lemma tells us about the powers of p dividing various cyclotomic polynomials.
Lemma 25.
Let p≠r be a prime and suppose that q is a power of r.
Writing d for the order of q modulo p,
p∣Φe(q) if and only if e=pad for some a≥0 (except if p=2 and d=2, where 2∣Φ1(q) as well).
If e is not the order of q modulo p, then p2∤Φe(q).
We have
Φd(qp)={Φd(q)⋅Φpd(q)if p∤d,Φpd(q)if p∣d.
Therefore for all d, the powers of p dividing Φd(qpa) and paΦd(q) are the same.
We now need information about the cross-characteristic Sylow structure of a group of Lie type, which is described in [7, Theorem 4.10.2]. We give a summary now, tailored to our needs.
Proposition 26.
Let G0=G0(q) denote a quasisimple group of Lie type, with Z(G0) a p′-group. Let d denote the order of q modulo p, and let pa be the exact power of p dividing Φd(q). Let P be a Sylow p-subgroup of G0.
There exists an abelian normal subgroup P0 of G0, of exponent pa, such that P/P0 is isomorphic to a subgroup of the Weyl group of G0, unless one of the following holds:
p=3,
G0=D43(q), where P0 has exponent pa+1,
p=2,
G0=G22(q), where P is elementary abelian of order
8.
Furthermore, if G is an almost simple group containing G0 as a normal subgroup, with G/G0 consisting of diagonal automorphisms, then the same results hold.
This means that, in the notation of the proposition, if the exponent of the Sylow p-subgroup of the Weyl group of G0 is pb, then the exponent of P is at most pa+b (except in the one case, where it is at most pa+b+1).
To get a bound for the maximal order of u, we finally need to consider the outer automorphism group of G0, which is more or less completely described in [7, Theorem 2.5.12]. Thus the contribution to o(u) comes from three sources: the toral contribution, the p-part of Φd(q) (except for D43), the Weyl contribution, the exponent of the Sylow p-subgroup of the Weyl group, and the outer contribution, the exponent of the Sylow p-subgroup of the outer automorphism group of G0. This is usually far greater than the actual maximal order of u, and so we use this to reduce the possible options for G0, and then use more explicit techniques to get better bounds on o(u) if required.
Notice that the Sylow p-subgroup of G0 is abelian if and only if p divides exactly one of the Φd(q) that divide |G0|, or in other words, if the Sylow p-subgroup of G0 is non-abelian and the order of q modulo p is d, then both Φd(q) and Φpd(q) divide |G0(q)|.
We also need to consider regular semisimple elements, in particular to know that their centralizer is abelian when u is a regular semisimple element in a cyclic Sylow subgroup. This result appears in [3, Proposition 9.1]. We remind the reader of the definition of a regular number. If W is a Coxeter group, a regular element is an element that acts regularly on the reflection representation, and a regular number is an number that is the order of a regular element. These are enumerated in [28, Section 5].
Lemma 27.
Let p be a prime and let q be a power of a prime r≠p, and let G(q) be a finite group of Lie type. Suppose that the order of q modulo p is a regular number, and that the Sylow p-subgroup of G(q) is cyclic. If u is a generator for the Sylow p-subgroup of G(q), then the centralizer CG(u) is abelian.
8 Classical groups in cross characteristic
In this section we consider the case where G0 is a central extension of a classical group in characteristic r≠p.
In [5], Di Martino and Zalesski solve the problem of which elements of quasisimple classical groups act minimally actively on the Weil modules (in fact, they do all almost cyclic elements). However, they only allow u to induce a inner-diagonal outer automorphism on G0 if it is linear or unitary, and only an inner automorphism if G0 is symplectic. The theorem in [5], applied to minimally active modules only (i.e., where u is a p-element and the characteristic of the field is p), is as follows.
Theorem 28.
Let G0 be one of SLn(q) (n≥3), SUn(q) or Sp2n(q), and let u either be in G0 or induce an inner-diagonal automorphism on G0 if G0 is not symplectic. Suppose that G0 is not one of the groups considered in Proposition 24. If u acts minimally actively on a Weil module, then one of the following holds:
G=Sp2n(q),
n is a power of
2,
pa=qn+12 for some a≥1,
o(u)=pa,
G=Sp2n(3),
n≠p is an odd prime and pa=3n-12 for some a≥1, with o(u)=pa,
G0=SUn(q),
n≠p is an odd prime at least
5,
pa=qn+1q+1 for some a≥1, and o(u)=pa,
G0=SLn(q),
n≠p is an odd prime, pa=qn-1q-1 for some a≥1, and o(u)=pa.
We will add to this by proving the following result.
Proposition 29.
Let G0 be a central extension of a simple special linear, unitary or symplectic group, but not one of the groups in Proposition 24. Let u∈G be a p-element, and let M be a simple module on which u acts minimally actively. If M↓G0 involves a Weil module, then u induces an inner-diagonal automorphism on G0.
We begin by proving, for G0 classical and not of type PSL2, that if M is not a Weil module, then the possibilities for an element of G acting minimally actively are very limited, restricted mostly to cases of exceptional Schur multipliers given in Proposition 24.
Table 4 is a summary of what we will need about the dimensions of Weil modules, and lower bounds for the dimensions of non-Weil modules for classical groups, assuming that G0 is not one of the groups in Proposition 24. (Let κn be 1 if p divides qn-1q-1 and 0 otherwise.)
Table 4
Minimal dimension of a non-trivial projective representation for simple classical groups.
| Group | Bound | Reference |
| SLn(q) (Weil) | qn-qq-1-κn | [11] |
| SLn(q) (Weil) | qn-1q-1 | [11] |
| SLn(q), n=3,4 (non-Weil) | (q-1)(qn-1-1)gcd(n,q-1) | [11] |
| SLn(q), n≥5 (non-Weil) | (qn-1-1)(qn-2-qq-1-κn-2) | [11] |
| SUn(q) (Weil) | qn+q(-1)nq+1 | [12] |
| SUn(q) (Weil) | qn-(-1)nq+1 | [12] |
| SU3(q) (non-Weil) | (q-1)(q2+3q+2)6 | [12] |
| SU4(q) (non-Weil) | (q2+1)(q2-q+1)2-1 | [12] |
| SUn(q), n≥5 (non-Weil) | qn-2(q-1)(qn-2-q)q+1 | [12] |
| Sp2n(q), q odd (Weil) | qn±12 | [9] |
| Sp2n(q), all q (non-Weil) | q(qn-1)(qn-1-1)2(q+1) | [9], [27] |
| Ω2n+1(q) | qn-1(qn-1-1) | [13] |
| Ω2n+(q) | qn-2(qn-1-1) | [21] |
| Ω2n-(q) | (qn-1+1)(qn-2-1) | [21] |
8.1 Groups SLn(q), n≥3
For this subsection we let G0 be a quotient of SLn(q) for n≥3, and we exclude the cases of PSL3(2)=PSL2(7), PSL4(2)=Alt8, and PSL3(4) which are considered in Propositions 16 and 24. Suppose that M is a non-trivial simple module, but not a Weil module. From Table 4, the dimension of M is at least
(qn-1-1)(qn-2-1q-1-1)
for n≥5.
Note that if u is a p-element of G, then, as we saw in Proposition 26 and the discussion afterwards, the order of u is bounded by a product of numbers: the exponent of the Sylow p-subgroup of the outer automorphism group (the outer contribution); the exponent of the Sylow p-subgroup of the Weyl group (only if the Sylow p-subgroup of G0 is non-abelian, the Weyl contribution); the p-part of Φd(q) (the toral contribution).
We will let G0 be a group G(qt) and assume that u induces an automorphism on G0 that projects onto a field automorphism of order t in Out(G0).
Proposition 30.
Suppose that G0 is a central extension of a special linear group PSLn(qt) for some n≥3, with (n,qt)≠(3,2),(3,4),(4,2), and let u be a p-element of G such that G=〈G0,u〉. If u acts minimally actively on a non-trivial simple module M that is not a Weil module, then G=PSL3(3), p=13, o(u)=13 and dim(M)=11,13.
Proof.
Let G0 be a central extension of PSLn(qt) for some n≥3 and t≥1, with the exclusions given above of PSL3(2), PSL3(4) and PSL4(2). First, let n=3,4, and note that if M is a non-Weil simple module, then
dim(M)≥(qt-1)(q(n-1)t-1)gcd(n,qt-1).
If the Sylow p-subgroup of SLn(qt) is abelian, then, in the notation introduced after Proposition 26, the Weyl contribution is 1, the toral contribution is at most Φd(qt), where d=1,…,n, and the outer contribution is t, since diagonal automorphisms are not of concern by Proposition 26. Thus in all cases, we have that o(u)≤Φ3(qt)⋅t, in fact o(u)≤Φ3(q)⋅t2 by Lemma 25. From [10, Theorem 4.1], α(u)≤n for p>2, and we have a lower bound for dim(M) from Table 4. The equation dim(M)≤α(u)⋅(o(u)-1) for minimally active M from Lemma 5
(iv) now yields
(qt-1)(q(n-1)t-1)gcd(n,qt-1)≤n((q2+q+1)t2-1);
since p∣t, we can assume that t is odd, and all solutions are for t=1, with q=2,3,4,7 for n=3 and q=2,3 for n=4. Removing those excluded from the start of this section, for t=1 we need consider (n,q)=(3,3),(3,7),(4,3). For PSL3(7), the only prime for which the Sylow p-subgroup is abelian is p=19, dim(M)≥96 (by Table 4) and α(u)≤3, so it in fact fails the bound. For PSL4(3), dim(M)≥26 (again, table 4) and p=5,13, with o(u)=p. Since α(u)≤4, this shows that p=5 cannot yield a minimally active module, and for p=13 we see that CG0(u)=Z(G0)⋅〈u〉, so that dim(M)≤p+1 for M to be minimally active by Lemma 8. Thus there are no examples here.
For PSL3(3) we have p=13, and again CG0(u)=〈u〉 (there are no central extensions) so we may apply Lemma 8. The simple modules from [16] have dimensions 1, 11, 13, 16 and 26, so those of dimensions 11 and 13 are minimally active.
Suppose that the Sylow p-subgroups are non-abelian, so that p=2,3. First assume p=3: the exponent of the Sylow 3-subgroup of the Weyl group is 3, and the toral contribution of o(u) is at most qt+1, in fact t(q+1) by Lemma 25. The outer contribution is at most t, so that o(u)≤3t2(q+1). Thus our equation dim(M)<α(u)⋅o(u) becomes
(qt-1)(q(n-1)t-1)gcd(n,qt-1)<3nt2(q+1);
which yields only t=1 and (n,q)=(3,2),(3,4),(4,2), all of which are excluded.
For p=2, we have a graph automorphism to consider as well. The toral contribution to o(u) is at most q+1, the Weyl contribution is at most n and the outer contribution if the lowest common multiple of 2 and t.
Hence α(u)⋅o(u) is at most nmt(q+1)⋅lcm(2,t), where m=n for n≥5, m=4 for n=3 and m=6 for n=4. We will check both Weil and non-Weil modules simultaneously, and all n, so we need
nmt(q+1)⋅lcm(2,t)≥α(u)⋅o(u)>dim(M)≥qnt-qtqt-1-1.
For n≥5 we only get (n,q,t)=(5,3,1). For n=4 we get (n,q,t)=(4,3,1), (4,5,1), and for n=3 we get qt≤23.
For PSL5(3), the exponent of the Sylow 2-subgroup is 16, so that o(u)≤32, α(u)≤5, and dim(M)≥120. Thus we need o(u)=32, and in this case v=u2 is an element of PSL5(3) of order 16. However, a simple computer check confirms that α(v)=2 for these elements, so that there is no minimally active module.
For PSL4(3) and PSL4(5), we have α(u)≤4 if 〈u〉∩G0, generated by v say, is non-trivial. Since the exponent of a Sylow 2-subgroup of both groups is 8, we get that o(u)≤16, and dim(M)≥26,124, respectively. This eliminates PSL4(5) as dim(M)>o(u)⋅α(u) (using Lemma 5
iv). For PSL4(3), if o(v)=8, then α(v)=2 and if o(v)=4, then α(v)≤3. Thus we can only get a minimal action of u on M if dim(M)=26 (the next smallest is dimension 38) and o(u)=16, with u therefore inducing the graph automorphism on M. However, there is no element of Aut(PSL4(3)) of order 16, as we see from [2, pp. 68–69]. Thus we get no minimally active modules here either.
Finally, consider n=3. If t=1, then α(u)≤3, and also the Weyl contribution to o(u) is 2, not n which is 3. Thus in this case we get o(u)⋅α(u) to be at most 12(q+1), which yields q≤13 for there to be a minimally active module. If we replace q+1 by the 2-part of q2-12, which is the actual toral contribution, we obtain q≤9. For these groups we check in [2] that the exponents of the Sylow 2-subgroups of Aut(PSL3(q)) are 8,8,16,16, as q=3,5,7,9, respectively. Since dim(M)≥12,30,56,90 from the table above, and α(u)≤3, we see that in fact G0=PSL3(3) is the only possibility. In this case, if u∈G0.2 has order 4 or 8, we use a computer to check that α(u)=2, and if u∈G0.2 has order 2, then α(u)≤4 by [10, Theorem 4.1]. Since if M is minimally active then dim(M)≤α(u)⋅(o(u)-1), we see that dim(M)≤14, so the 12-dimensional simple module is the only possibility, with o(u)=8 (the simple modules for G0 have dimensions 1, 12, 16 and 26). However, both classes of elements of order 8 in G0.2 have type (8,4) on M of dimension 12, so it is not minimally active.
Thus let n≥5 and p be odd. If M is not a Weil module, then
dim(M)≥(q(n-1)t-1)(q(n-2)t-qtqt-1-1),
and for all u, α(u)≤n. If the Sylow p-subgroup of G0 is abelian, then o(u) is at most (qnt-1)/(q-1)⋅t, and using the formula dim(M)<o(u)⋅α(u) yields only PSL5(2). As dim(M)≥75 for non-Weil modules from the formula above, and the order of prime-power elements of PSL5(2) is at most 8 or 31, we see that o(u)=31=Φ5(2). But in this case CG0(u)=〈u〉, so that dim(M)≤p+1=32, and there are therefore no examples.
Thus the Sylow p-subgroup of G0 is non-abelian, and therefore p divides two separate Φd-tori: from Lemma 25, we see therefore that if qt has order d modulo p, dp≤n, and therefore the toral contribution to u is at most qnt/p-1, with the Weyl contribution at most n and the outer contribution t. We therefore have that o(u)≤nt(q⌊nt/3⌋-1) (as p≥3), and using the formula yields
n2t(q⌊nt/3⌋-1)>(q(n-1)t-1)(q(n-2)t-qtqt-1-1),
which has no solutions for n≥5. This completes the proof.
∎
Having determined which non-Weil modules can be minimally active, we turn our attention to the Weil modules for odd primes p, where u induces a non-diagonal outer automorphism, which must involve a field automorphism of order at least 3.
As in the proof of the previous proposition, the order of an element of G0 is at most either t(qn-1) or tn(q⌊n/3⌋-1), and we multiply this by the outer contribution, which is t, and α(u), which is at most n, to get an estimate for o(u)⋅α(u). We then compare that to (qnt-qt)/(qt-1) for t≥3, and find only one possible solution: PSL3(8).3, of course with p=3. In this case, dim(M)≥72 by [16, p. 187], and o(u)≤9 by [2, p. 74], so this cannot work either.
Thus if u∈G acts minimally actively on a Weil module, then it induces a inner-diagonal automorphism on G0, proving Proposition 29 for linear groups.
8.2 Groups SUn(q)
This looks very similar to the linear case in the previous subsection. We start by dealing with non-Weil representations, with the cases n=3,4 having to be dealt with separately, and at the same time proving that there are no minimally active modules for p=2, Weil or non-Weil. This then allows us to prove easily that u cannot act minimally actively on a simple module without inducing an inner-diagonal automorphism on G0, just as with the linear case.
Note that we exclude PSUn(q) for (n,q)=(3,2),(4,2),(4,3),(6,2). For PSU3(3)=G2(2)′, we require p≠2,3.
Proposition 31.
Suppose that G0 is a central extension of a special unitary group PSUn(qt) for some n≥3, with (n,qt)≠(3,2),(4,2),(4,3),(6,2), and let u be a p-element of G such that G=〈G0,u〉. If u acts minimally actively on a non-trivial simple module M, then M is a Weil module.
Proof.
Let G0 be a central extension of PSUn(qt), excluding the groups listed above the proposition, and let M denote a non-Weil simple module. We first consider the case n=3, where dim(M)≥16(qt-1)(q2t+3qt+1), and p is either 2 or 3, or divides one of qt-1, qt+1, or q2t-qt+1=Φ6(qt). We apply Lemma 25 to replace Φd(qt) by tΦd(q), as in the case for PSLn(qt). Unless qt=3 and p=2, from [10, Theorem 4.1] we have that α(u)≤3.
If p≠2,3, we have that o(u)≤t2Φd(q), with the Weyl contribution being trivial, the toral contribution being tΦd(q) for d=1,2,6, and the outer contribution being t. (Note that diagonal automorphisms need not be considered, as in PSLn(qt), using Proposition 26.)
If d=1,2, then we only end up with PSU3(3), and if d=6, then since this is a regular number CG0(u) is abelian by Lemma 27, and hence if M is minimally active, then dim(M)≤2o(u) by Lemma 8. This forces t=1 and q<7, so only q=3,5, since q=2,4 are excluded. For q=3,5, q2-q+1=7,21, and so p=7 in both cases. However, this now excludes q=5, leaving only q=3. Here, if M is not a Weil module, then dim(M)≥14, and there are therefore no examples.
If p=3, then the toral contribution is at most q+1, the Weyl contribution is 3 and the outer contribution is t, so we place this in our formula to get only (n,q,t)=(3,2,3),(3,5,1). The exponents of the Sylow 3-subgroups of the automorphism group Aut(PSU3(qt)) for qt=5,8 are 3 and 9, respectively, whereas dim(M)≥20,56, respectively, from [16]. Thus there are no minimally active modules for p=3.
Suppose that p=2. The toral contribution is at most t(q+1), the Weyl contribution is at most n, and the outer contribution is at most 2t. Unless n=4 or G0=PSU3(3), we have α(u)≤n. Thus α(u)⋅o(u)≤2t2n2(q+1), and for this to be at least dim(M) (for any non-trivial M, not just non-Weil modules), for n≥5 we have that (n,q,t)=(5,3,1),(6,3,1). In these two cases, dim(M)≥60,182, whereas the exponent of the Sylow 2-subgroup of Aut(PSU5(3)) is 16, and the exponent of the Sylow 2-subgroup of PSU6(3) is 16, so that of the automorphism group is at most 32. If v∈PSUn(3) for n=5,6 has order 8 or 16, then α(v)≤3, so that M cannot be minimally active.
If n=4, then α(u)≤6, and this yields q=3,5,7 for t=1, and q=3 for t=2. For q=3,5,7,9, dim(M)≥20,104,300,656, with the exponents of the Sylow 2-subgroups of G0 being 8,8,16,16, respectively. Thus only q=3 can yield a minimally active module, but PSU4(3) is excluded from consideration.
For n=3 we need better bounds, because for t=1 we get q≤19 satisfying the bound
2t2n2(q+1)≥qnt-qtqt+1,
and for t=2 we get q=3. The Weyl contribution (n in the above inequality) may be replaced by 2, and the toral contribution (t(q+1) above) may be replaced by the 2-part of 12(q2t-1). Doing so yields qt≤9, and replacing o(u) with the correct exponents, which are 8,8,16,16 for qt=3,5,7,9, respectively, means that qt=9 can be excluded, as dim(M)≥6,20,42,72. If α(u)=2 for u of maximal order, then this will exclude qt=5,7 as well: this can be checked and is indeed the case, yielding G0=PSU3(3)=G2(2)′, so already considered. This completes the proof for p=2, all simple modules and all n≥3.
We now let n=4, and now p is odd. If p>3, then the Sylow p-subgroup is abelian, so d=1,2,4,6, and the toral contribution is at most t(q2+1); the Weyl contribution is 1; and the outer contribution is t. Since α(u)≤n and dim(M)≥12(q2t+1)(q2t-qt+1)-1, plugged into α(u)⋅o(u)>dim(M) yields
nt2(q2+1)>12(q2t+1)(q2t-qt+1)-1,
which yields t=1 and q=2,3, both of which are excluded from consideration.
If p=3, then the Weyl contribution is 3 and the toral contribution is at most t(q+1), with t the outer contribution, yielding
3nt2(q+1)>12(q2t+1)(q2t-qt+1)-1.
Again, only qt=2,3 satisfy this, which have been excluded.
Thus n≥5. First suppose that the Sylow p-subgroup is abelian. The toral contribution is at most t(qn-1)/(q-1) (as this is greater than t(qn+1)/(q+1), and the order d of qt modulo p is either at most n or 2m for some odd m≤n), the Weyl contribution is 1, and the outer contribution is at most t. For M a non-Weil module, the inequality α(u)⋅o(u)<dim(M) becomes
nt2(qn-1)q-1>q(n-2)t(qt-1)(q(n-2)t-qt)qt+1,
and this forces t=1 and (n,q)=(5,2),(5,3),(6,2),(7,2),(8,2). To eliminate these, we first ignore the ts, and then produce better estimates for the toral contribution than (qn-1)/(q-1): for n=5,7 we get (qn+1)/(q+1), which eliminates (5,3) and (7,2), and for n=8 we use (q7+1)/(q+1), which eliminates (8,3). Since (6,2) is not being considered in this proposition, we are left with PSU5(2). Here p=5,11, dim(M)≥43 for non-Weil modules from [16, pp. 182–184], and it is easy to see that α(u)=2 for o(u)=5,11 by a computer check (alternatively we can use the fact that CG(u) is abelian and apply Lemma 8). Thus there are no minimally active non-Weil modules in this case.
If the Sylow p-subgroup is non-abelian, then, as with the linear case, the toral contribution is at most t(q⌊n/3⌋+1) and the Weyl contribution is at most n, yielding
n2t2(q⌊n/3⌋+1)>q(n-2)t(qt-1)(q(n-2)t-qt)qt+1,
where we get t=1 and (n,q)=(5,2),(6,2), although PSU6(2) is excluded. For G0=PSU5(2), only the Sylow 3-subgroup (and the Sylow 2-subgroup of course) is non-abelian, and Out(G0) has order 2, we have G0=G and so u has order at most 9, and α(u)=2 if o(u)=9, and α(u)≤5 if o(u)=3, with dim(M)≥44 from [16, p. 181]. Thus there is no non-Weil simple minimally active module for this group.
∎
As with linear groups, we now check that if u induces an automorphism that is not inner-diagonal on G0, then u does not act minimally actively on a Weil module. From the previous proposition we may assume that p is odd, so that t≥3.
Suppose that the Sylow p-subgroup of G0 is abelian: as in the proof of the proposition we see that o(u)≤t2(qn-1)/(q-1), and so we get
nt2(qn-1)q-1>qnt-qtqt+1,
yielding (n,q)=(3,2),(4,2) for t=3, and no solutions for t≥5. Thus p=3, but the Sylow 3-subgroup of G0 is definitely not abelian.
If the Sylow p-subgroup of G0 is non-abelian, then the toral contribution is at most t(q⌊n/3⌋+1), the Weyl contribution is at most n, and this time we get
n2t2(q⌊n/3⌋+1)>qnt-qtqt+1,
and this yields (n,q,t)=(3,2,3) as the only solution, so again p=3. The Sylow 3-subgroup of Aut(PSU3(8)) has exponent 9, and the dimension of a Weil module is 56, with α(u)≤3 by [10, Theorem 4.1], so u cannot act minimally actively on a Weil module by Lemma 5
(iv).
This completes the proof of Proposition 29 for unitary groups.
8.3 Groups Sp2n(q)
For this subsection we let G0 be a quotient of Sp2n(qt), and we exclude the cases of Sp4(2)=Sym6, PSp4(3)=PSU4(2) and Sp6(2) (the last two appear in Proposition 24). Suppose that M is a non-trivial simple module, but not a Weil module, which exist only for odd q. From the table near the start of this section, the dimension of M is at least qt(qnt-1)(q(n-1)t-1)/2(qt+1).
Proposition 32.
Suppose that G0 is a central extension of a symplectic group PSp2n(qt) for some n≥2, with (n,qt)≠(4,2),(4,3),(6,2), and let u be a p-element of G such that G=〈G0,u〉. If u acts minimally actively on a non-trivial simple module M that is not a Weil module, then G=Sp4(4), p=17, o(u)=17, and dim(M)=18.
Proof.
Let G0=Sp2n(qt) for some n≥2, some prime power q and some t≥1. As in previous sections, u induces an automorphism that projects in Out(G0) to a field automorphism of order t. Since M is not a Weil module, we have
dim(M)≥(qnt-1)(qnt-qt)2(qt+1).
Let d be the order of qt modulo p.
If d=2n, then u is regular, so dim(M)<2o(u) by Lemma 27. We have that o(u)≤t2(qn+1) by Lemma 25, since the outer contribution is t and the toral contribution is at most t(qn+1), so we get
2t2(qn+1)≥(qnt-1)(qnt-qt)2(qt+1).
If t=1, then the solutions to this are q≤5 for n=2, and q=2 for n=3. If t>1, we only get (n,q,t)=(2,2,2), but this needs p∣t=2 and p∤q=4, a contradiction. We of course exclude (n,q)=(2,2),(2,3),(3,2) for t=1, as we stated above, so we are left with (n,q)=(2,4),(2,5) for t=1. Here d=4, and Φd(q)=q2+1: 42+1=17 and 52+1=26. For q=5 this means that o(u)=13, and dim(M)≥40, so this cannot work, but for Sp4(4), the module of dimension 18 could be minimally active. Since CG(u)=〈u〉 in this case, it is minimally active by Lemma 8.
Suppose that the Sylow p-subgroup is abelian. If d≠n, then we have the bound o(u)≤t2(qd+1) with d≤n-1, and note that n≥3. We also have that α(u)≤n+3 by [10, Theorem 4.3], so using Lemma 5
(iv) we get
(n+3)⋅t2(qd+1)≥α(u)o(u)≥dim(M)≥(qnt-1)(qnt-qt)2(qt+1),
and the only solutions are for t=1, with (n,q)=(3,2),(4,2). The first can be ignored using Proposition 24, and for the second we have d=1,2,3,4,6, which yield Φd(2)=1,3,7,5,3. Since the Sylow 3-subgroup of Sp8(2) is non-abelian, we only get the cases d=3,4, so p=7,5. For p=7 it is easy to check with a computer that α(u)=2, o(u)=7, and dim(M)≥35 by the degree bound above. Thus there is no (non-trivial) minimally active simple module for this group. For p=5 we have o(u)=5 and α(u)≤3, so that again there are no examples.
We need also consider the case where d=n, so that o(u)≤t2(qn-1)/(q-1). Using α(u)≤(n+3) and Lemma 5
(iv), we get that if M is minimally active, then
(n+3)⋅t2(qn-1)q-1≥α(u)o(u)>dim(M)≥(qnt-1)(qnt-qt)2(qt+1),
which for t=1 yields the solutions (n,q)=(2,2), (2,3), (2,4), (3,2), (4,2), (5,2), and for t≥2 yields only the solution (n,q)=(2,2) for t=2, which we noticed earlier is not an example because p∣t=2 and p∤q=4. If n is odd, then n is a regular number, so we may replace n+3 by 2, as in the previous case, and this removes the case (5,2). The cases (2,2),(2,3),(3,2) are excluded from our analysis, and (4,2) has been dealt with, leaving only Sp4(4), with p∣Φ2(4)=5. We have dim(M)≥18, o(u)=5 and α(u)≤3 by a computer calculation, so there is no example here either.
We may therefore assume that the Sylow p-subgroup is non-abelian, and hence p divides the order of the Weyl group of type C, which is Z2≀Symn, and p≤n with p dividing two separate tori, so that p∣Φd(q) and p∣Φpd(q). If p is odd, then in particular this means that d≤n3 or d is even and d≤2n3, so in either case the toral contribution is at most t(qd+1). The Weyl contribution is at most n, and the outer contribution is t, so o(u)≤nt2(qd+1). This is of course maximized at d=⌊n3⌋. As α(u) is still at most n+3, we get
(n+3)⋅nt2(q⌊n/3⌋+1)≥α(u)o(u)>dim(M)≥(qnt-1)(qnt-qt)2(qt+1),
The only solution is (n,q,t)=(3,2,1), which we have already excluded.
Thus p=2. Here there is a diagonal automorphism, but we do not need to consider these by Proposition 26, so the outer contribution is t. The Weyl contribution is 2n, and the toral contribution is at most t(q+1) by Lemma 25, so the order of a 2-element of G0 is at most 2nt2(q+1) by Proposition 26. Since α(u)≤2n, this yields α(u)⋅(o(u)-1)≤2n(2nt2(q+1)-1).
If M is minimally active, then, by Lemma 5
(iv)
2n(2nt2(q+1)-1)≥α(u)(o(u)-1)≥dim(M)≥(qnt-1)(qnt-qt)2(qt+1),
which has no solutions for t≠1, and for t=1 we get (n,q)=(2,2), (2,3), (2,5), (3,2), with the first and last eliminated since q must be odd. Since Sp4(3) is also excluded, this leaves G0=Sp4(5): from [2, p. 63] we see that o(u)≤8, and α(u)≤4 with dim(M)≥40, so M cannot be minimally active.
This completes the proof.
∎
We now have to complete the proof of Proposition 29 by checking that if u induces an outer automorphism on G0=Sp2n(qt), then u cannot act minimally actively on a Weil module. First suppose that p is odd, so that u induces a field automorphism and t≥3.
The dimension of M is 12(qnt±1) (recall that q must be odd) and as we saw above, if the Sylow p-subgroup of G0 is abelian, then o(u)≤t2(qn+1), and as α(u)≤n+3, if u acts minimally actively, then by Lemma 5
(iv) we have
t2(n+3)(qn+1)≥12(qnt-1),
which yields (n,q,t)=(2,3,3), but of course t, which is a power of p, cannot divide q, so we get no examples.
If the Sylow p-subgroup is non-abelian, then
α(u)o(u)≤(n+3)nt2(q⌊n/3⌋+1),
as we saw in the proof of the previous proposition: thus we have
(n+3)nt2(q⌊n/3⌋+1)≥12(qnt-1),
and this has no solutions.
We thus reduce to p=2. In this case, from [9, Section 5], we see that there are two Weil modules, which have dimension 12(qn-1) and are swapped by the diagonal automorphism. By Theorem 28, v∈G0 cannot act on these Weil modules with a single Jordan block, and hence by Lemma 9 if u induces a diagonal automorphism on G0, then it cannot act minimally actively on the sum of the two Weil modules.
Thus u acts as either a field automorphism or the product of a field and diagonal (whichever stabilizes the two Weil modules), but in either case t≥2.
We have already bounded o(u) by 2nt2(q+1), so with α(u)≤2n we get
2n(2nt2(q+1)-1)≥α(u)⋅(o(u)-1)≥dim(M)≥12(qnt-1).
If t≥4, then there are no solutions, and for t=2 we get solutions (n,q)=(2,3), (2,5), (3,3). The exponents of the Sylow 2-subgroups of PSp4(9), PSp4(25) and PSp6(9) are 8,8,16, respectively, so o(u)≤16,16,32, respectively. The dimensions of the Weil modules are 40, 312, 364, respectively, and α(u)≤4,4,6, respectively, so the formula dim(M)<α(u)⋅o(u) eliminates the second and third options from being minimally active. Finally, for PSp4(9), if we can reduce α(u) for u of order 16 (hence v=u2∈G0 of order 8) to 2, then we are done: this is the case by an easy computer calculation, and we complete the proof of Proposition 29.
8.4 Groups Ω2n+1(q) and Ω2n±(q)
As we saw in Table 4, the minimal degree for Spin2n+1(q) for (n,q)≠(3,3) is qn-1(qn-1-1).
Recall that the polynomial order of Spin2n+1(q) is
qn2∏i=1n(q2i-1),
so that if p∤q divides the order of Spin2n+1(q), p divides qd±1 for some 1≤d≤n.
Proposition 33.
Let G0 be a central extension of one of the groups Ω2n+1(q) for (n,q)≠(3,3) and n≥3. Let u be a p-element of G such that G=〈G0,u〉. There are no non-trivial minimally active simple modules for G.
Proof.
Let G0=Ω2n+1(qt) for some n≥3, some prime power q, and some t≥1, and suppose that the Sylow p-subgroup of G0 is abelian, so that p divides a single cyclotomic polynomial, and let d be the order of qt modulo p. The order of u is at most t⋅tΦd(q)≤t2⋅(qn+1). As α(u)≤n+3 by [10, Theorem 4.4], we get using Lemma 5
(iv)
(n+3)⋅t2(qn+1)>α(u)⋅o(u)>qt(n-1)(qt(n-1)-1)
if M is minimally active, and this forces t=1 and (n,q)=(3,3),(3,5). Omitting the t from now on, replacing the upper bound qn+1 for Φd(q) by each of (qn+1)/(q+1), (qn-1)/(q-1), (q-1) and (q+1) eliminates (n,q)=(3,5), and (n,q)=(3,3) is excluded already, so there are no solutions.
We now may assume that the Sylow p-subgroup is non-abelian, so that if p is odd then o(u)≤t⋅n⋅t(qd+1) (as the Weyl group of type B is the Weyl group of type C we can use the Weyl contribution from Proposition 32), but with both d and pd dividing 2n. We thus get
(n+3)⋅n⋅t2(qd+1)>α(u)⋅o(u)>qt(n-1)(qt(n-1)-1),
for d≤n3, which is obviously maximized at d=⌊n/3⌋, still with no solutions.
If p=2, then we get o(u)≤2t⋅2n⋅t(q+1) using Proposition 26 and the fact that the order of an outer automorphism is at most 2t, and so now we have
2n⋅4nt2(q+1)>α(u)⋅o(u)>qt(n-1)(qt(n-1)-1),
(as α(u)≤2n this time) which again has no solutions for (n,q,t)≠(3,3,1). This completes the proof.
∎
Having dispensed with the odd-dimensional orthogonal groups, we turn to the even-dimensional ones. For Ω2n+(q), the minimal degree is qn-2(qn-1-1) (unless G0=Ω8+(2)) and for Ω2n-(q) the minimal degree is (qn-2-1)(qn-1+1), so in both cases dim(M)>(qn-1-1)(qn-2-1). If we use this bound, then we can deal with both cases simultaneously. The polynomial order of Spin2n±(q) is
qn(n-1)(qn∓1)∏i=1n-1(q2i-1).
We already found minimally active modules for 2⋅Ω8+(2) in Proposition 24, and the next proposition says that there are no more.
Proposition 34.
Let G0 be a central extension of one of the groups Ω2n±(q) other than Ω8+(2), for n≥4. Let u be a p-element of G such that G=〈G0,u〉. There are no non-trivial minimally active simple modules for G.
Proof.
Our proof works the same as Proposition 33. If the Sylow p-subgroup is abelian, then p is odd and o(u)≤t2Φd(q)≤t2(qn+1). Placing this in our standard formula from Lemma 5
(iv), using α(u)≤n+3 from [10, Theorem 4.4] gives
(n+3)t2(qn+1)≥dim(M)>(qt(n-1)-1)(qt(n-2)-1).
This bound yields no solutions for t≥3, and for t=1 the solutions
(n,q)=(4,2),(4,3),(4,4),(4,5),(4,7),(5,2),(6,2).
If q is odd, then we may replace (qn+1) by 12(qn+1), removing (4,5) and (4,7) from the list.
If d=n or d=2n, then u is regular, so CG0(u) is abelian by Lemma 27, and so in this case we may replace α(u)⋅(o(u)-1) by 2o(u) via Lemma 8, and so (removing the t, which is equal to 1 anyway)
2(qn+1)>(qn-1-1)(qn-2-1),
which only has a solution for (n,q)=(4,2). Thus we may assume that d≠n,2n, in which case we may replace o(u) by t2(qn-1+1). Using this we reduce our possibilities to (4,2) and (5,2).
As Ω8+(2) is excluded, we just consider G0=Ω8-(2): from [16] we see that dim(M)≥33, and o(u)=3,5,7,9,17. Furthermore, we have that α(u)=2 for o(u)>3, and α(u)≤4 for o(u)=3, so there are no examples using the formula dim(M)≤α(u)⋅(o(u)-1).
For G0 a central extension of Ω10±(2), dim(M) is at least the smallest of 25-2(25-1-1)=120 and (25-1+1)(25-2-1)=119, so dim(M)≥119. For Ω10+(2), o(u)∈{3,5,7,9,17,31}, and for o(u)≥7 we have that α(u)=2, with α(u)≤3 for o(u)=5 and α(u)≤5 for o(u)=3, which shows that u cannot act minimally actively on a non-trivial M. For Ω10-(2), o(u)∈{3,5,7,9,11,17}, and the same statements hold for α(u), so again u cannot act minimally actively on a non-trivial M.
Suppose that p is still odd, but that the Sylow p-subgroup is non-abelian. Thus p divides both Φd(qt) and Φdp(qt), and d≤n3. Since the Weyl group of type D is a subgroup of the Weyl group of type B, we see that the exponent of the Sylow p-subgroup of the Weyl group is at most n. Thus o(u)≤t2⋅n⋅(q⌊n/3⌋+1) and α(u)≤n+3, and thus we need to check
t2n(n+3)(q⌊n/3⌋+1)≥(qt(n-1)-1)(qt(n-2)-1),
which only has solutions for t=1, and then (n,q)=(4,2),(5,2), which we have already checked. This completes the proof for p odd.
Suppose that p=2, so that the order of u is at most 4t2⋅2n⋅(q+1): Out(G0) has exponent at most 4t, the Weyl contribution is at most 2n, and the toral contribution is at most t(q+1). (To see that the exponent of Sylow 2-subgroup of Out(G0) is at most 4t and not 8t, note that if n is even, then the diagonal automorphisms form Z2×Z2, so we are done, and if n is odd, then the diagonal automorphisms form either Z2 or Z4, with the graph automorphism inverting this group [7, Theorem 2.5.12 (i)].) Since α(u)≤2n, we get
16t2n2(q+1)≥(qt(n-1)-1)(qt(n-2)-1),
which has a solution only for (n,q,t)=(4,3,1). However, although information on the Sylow 2-subgroups of Aut(Ω8±(3)) is not available in [2], there are constructions of them on the online Atlas, and hence a computer algebra package immediately tells you that the exponent is 8, not 32 as suggested by the formula above. This proves that there are no minimally active modules for p=2.
∎
8.5 Groups SL2(q)
This short subsection deals with the groups G0=SL2(q), where p∤q, q≥4 and q≠4,5,7,9 (as these are alternating groups or are given in Proposition 24). In [5, Theorem 1.2], if SL2(q)≤G≤GL2(q), then all possibilities for u acting minimally actively are determined, and given by the following lemma.
Lemma 35.
Let G0 be a central extension of PSL2(q) for q≠4,5,7,9. Suppose that u induces an inner-diagonal automorphism on G0. If u acts minimally actively on M, then one of the following holds:
G=SL2(q) for q=2a,
p=2a±1 is a Fermat or Mersenne prime, o(u)=p,
M is any simple module,
G=SL2(q) for q odd, p is odd, 12(q±1)=pa,
o(u)=pa,
dim(M)≤o(u)+1,
G=PSL2(q) or PGL2(q),
q is a Fermat or Mersenne prime, p=2, and dim(M)≤o(u)+1.
Thus we can assume that u∈G acts as a field or product of a field and diagonal automorphism. Let G0 be a central extension of PSL2(qt) for some prime power q and some t≥2 a power of p, with G=〈G0,u〉 and |G:G0|=t. Note that 12(qt-1) is the smallest dimension of a non-trivial simple module for G0 if q is odd, and qt-1 is the smallest dimension if q is even.
Suppose that 〈u〉∩G0=1. If o(u) is even, then α(u)≤4 by [10, Lemma 3.1], so dim(M)≤4(t-1) by Lemma 5
(iv). This yields 12(qt-1)≤4(t-1), so t=2 and qt≤9. Having excluded 4,5,7,9, and since 8 need not be considered, there are no solutions.
If o(u) is odd, then as α(u)=2 by [10, Lemma 3.1], we see that dim(M)≤2(t-1), and dim(M)≥12(qt-1). As t≥3, we only get qt=8, but then the minimal degree is qt-1, not 12(qt-1), and so there are no solutions here either.
Thus ut≠1, and so p∣(q±1). If p=2, the order of u is at most t2(q+1), and α(u)=3 by [10, Lemma 3.1], so if u acts minimally actively on M, then we have dim(M)<32t2(q+1), whereas dim(M)≥12(qt-1). For t=2 this yields q≤11, for t=4 this yields q=3, and there are no solutions for t≥8.
Replacing t(q+1) with the 2-part of 14(q2t-1) (which is the exponent of the Sylow 2-subgroup of G0) yields q≤9.
Finally, in the remaining cases, one may check that the exponent of the Sylow 2-subgroup of Aut(PSL2(qt)) is 8, 16 and 16, for qt=25,49,81, respectively, and dim(M)≥12,24,40, respectively. This eliminates the case where qt=81. Finally, in the remaining cases if dim(M)≥qt-1, then it cannot be minimally active, so it is only the two modules of dimension 12(qt-1) that are important: for these, a computer calculation shows that any element v of G0 acts on them with only blocks of size o(v), whence u cannot act minimally actively.
Thus p is odd. First assume that q is even. The order of p is at most 12t2(q+1), and α(u)=2 by [10, Lemma 3.1], so that if u acts minimally actively, then we have dim(M)≤t2(q+1), whereas dim(M)≥qt-1. The only solution to this is q=2 and t=p=3, which is the small Ree group G22(3), hence will not be considered as it is defining characteristic.
Hence we may assume that q is odd, in which case dim(M)≥12(qt-1). We still have that α(u)=2 and o(u)≤12t2(q+1)/2. This yields only one solution again, namely q=3 and t=3, but then this is defining characteristic and not in consideration. Thus there are no solutions when u does not induce an inner-diagonal automorphism.
9 Exceptional groups in cross characteristic
In this section we deal with G0 a central extension of an exceptional group of Lie type. We start by dealing with a few small groups, which feature because they have exceptional Schur multipliers and so can have unusually small minimal faithful degrees, the analogue of Proposition 24.
Proposition 36.
Let G0 be a central extension of one of the simple groups G2(3), G2(4), F4(2), B22(8). Let u be a p-element of G such that G=〈G0,u〉, and let M be a non-trivial simple module on which u acts minimally actively. One of the following holds:
G=G2(3),
p=13,
o(u)=13 and dim(M)=14,
G=2⋅G2(4),
p=13,
o(u)=13 and dim(M)=12,
G=2⋅B22(8),
p=13,
o(u)=13 and dim(M)=14 (two representations).
The field of definition is always F13.
Proof.
The group G2(3) has outer automorphism group of order 2 and Schur multiplier of order 3, with the outer automorphism of 3⋅G2(3) inverting the centre, so we only have to consider the groups G=G2(3), G=3⋅G2(3) and G=G2(3).2. We simply check these one by one for p=2,7,13, and get the single example above.
The group G2(4) has outer automorphism group of order 2 and Schur multiplier of order 2, so we need to consider G2(4) and 2⋅G2(4) for p odd (so p=3,5,7,13), where the minimal degree is 12. The order of u must be p, and for p=5,7,13 two conjugates of u generate G, so we can exclude p=3,5 by Lemma 5
(iv) and focus on the 12-dimensional simple module for 2⋅G2(4) for p=7,13. As the centralizer of an element of order 7 in G2(4) has order 21, we get dim(M)≤10 by Lemma 8 and so can exclude this as well, leaving just p=13, where for G2(4) the centralizer has order exactly 13, so an application of the same lemma shows that the simple module of dimension 12 is minimally active for u.
For the group G=F4(2), there is an exceptional Schur multiplier of order 2: for G=2⋅F4(2), the character degrees are known for p=5,7,13,17, but the full set of character degrees is not known for p=3. For p≥5, the minimal faithful degree is 52, and for p=3 (see [27]) states that it is at least 44. A computer calculation shows that G is generated by two conjugates of u for o(u)=5,7,9,13,17, and so there are no minimally active modules for these elements. For o(u)=3, three conjugates suffice, and so there are no minimally active modules here either.
Now let G0 be a central extension of the group B22(8), where the exceptional Schur multiplier is a Klein four group, but all extensions 2⋅B22(8) are isomorphic because the outer automorphism of order 3 permutes them. Here p=5,7,13, as we can discount p=3, since o(u)=3 and α(u)≤3 by [10, Proposition 5.8], and dim(M)≥14. Thus we just check all simple modules for G=2⋅B22(8) and o(u)=5,7,13, noting that M is minimally active if and only if dim(M)≤p+1 because CG0(u)=Z(G0)⋅〈u〉 via Lemma 8. We find just the one example for p=13.
∎
In [10, Theorem 5.1], it is shown that α(G)≤ℓ+3 if G is of exceptional type, where ℓ is the untwisted rank of G (with one exception of G of type F4 and p=2, where α(G)≤8).
We also need a bound on the orders of p-elements in an exceptional group of Lie type. Broadly speaking, by Proposition 26 if q has order d modulo p, then the order of a p-element is at most qd-1 multiplied by the exponent of the Weyl group. We give a general bound now.
Proposition 37.
Let G0 be a central extension of one of G2(q), F4(q), E6(q), E62(q), E7(q) and E8(q) other than G2(2), with p∤q, and let u be a p-element of G. The order u is at most qℓ+1-1, where ℓ is the untwisted rank of G.
Proof.
Let u be a p-element, and let d be the order of q modulo p. If p does not divide the order of the Weyl group, then p≥5 (p≥7 for E8(q)) and the Sylow p-subgroup of the socle of G is abelian and has exponent at most Φd(q) by Proposition 26. The only outer automorphism of G that u can induce is a field automorphism, as diagonal and graph automorphisms have order 2 or 3, and field automorphism have order less than q-1.
Thus o(u)≤Φd(q)⋅(q-1). All we therefore need is an upper estimate for all Φd(q) where d divides one of the reflection degrees for G.
For G=G2, the largest is Φ3(q)=q2+q+1, so o(u)≤(q3-1). For F4, the largest is Φ8(q)=q4+1, so o(u)≤(q4+1)(q-1)≤q5-1. For E6 the largest is Φ9(q)=q6+q3+1, so o(u)≤(q6+q3+1)(q-1)≤(q7-1). For E62(q) the largest is Φ18(q)=q6-q3+1, so again we have o(u)≤q7-1. For E7 the largest is Φ7(q)=q6+q5+q4+q3+q2+q+1, so o(u)≤q7-1 again. Finally, for E8 the largest is Φ30(q)=q8+q7-q5-q4-q3+q+1, so that o(u)≤Φ30(q)(q-1)≤q9-1. Thus in all cases,
o(u)≤qℓ+1-1.
Thus now p divides the order of the Weyl group, so p≤7, and we use Proposition 26 again. The exponents of the Sylow p-subgroups of the Weyl groups of exceptional groups are given below.
| Group | Exponents |
| G2 | 2, 3 |
| F4 | 8, 3 |
| E6 | 8, 9, 5 |
| E7 | 8, 9, 5, 7 |
| E8 | 8, 9, 5, 7 |
For p=7, we have G=E7,E8 and either the Sylow p-subgroup is abelian, hence we are done by above, or p∣(q±1), whence the order of u is at most 7(q±1)(q-1)≤7(q2-1), where the Weyl contribution is 7, the toral contribution is (q±1), and the outer contribution is at most (q-1). The product of these is clearly less than q8-1.
For p=5, we have that the order of u is at most 5(q2+1)(q-1)≤5(q3-1), using the same argument, as d=1,2,4. Of course, this is still smaller than q7-1, which is the required bound as G=E6,E7,E8.
For p=3, the order of u depends on which group we are in. The toral contribution is at most q+1, the Weyl contribution is 3 for G2,F4 and 9 otherwise, and the outer contribution is at most 3(q-1) (diagonal for E6ε and field automorphisms). For G=G2,F4 we get 3(q2-1), which is at most (q3-1) for G2(q) (as q≠2,3), and at most q5-1 for all q≥2, so we get at most qℓ+1-1. For G=E6,E7,E8, we get 27(q2-1), which is less than (q7-1) for all q≥2, so in all cases again qℓ+1-1 will do.
For p=2, we get that the toral contribution is at most q+1, the Weyl contribution is 2 for G2 and 8 for all other groups, and the outer contribution is at most (q-1) for G2, F4 and E8, 2(q+1) for E6ε, and (q-1) for E7, by [7, Theorem 2.5.12], yielding at most 2(q2-1) for G2, which is less than q3-1 for q≥3.
For F4 we get o(u)≤8(q+1)≤(q5-1), for E6ε we have 16(q2-1)(q+1)≤(q7-1) for all q≥3, and for E7,E8 we have o(u)≤16(q+1)(q-1)≤(q8-1), as needed.
∎
Now we know that every p-element in G has order at most qℓ+1-1, then we get that dim(M)<α(G)⋅(qℓ+1-1), and we can apply the Landazuri–Seitz–Zalesskii bounds from [21] and [27].
| | | | α(G)⋅(qℓ+1-1) |
| Group | Landazuri–Seitz | LS evaluated | α(G) | evaluated |
| F4 (q≥4 even) | q72(q3-1)(q-1) | 448 (q=4) | ≤8 | ≤248 (q=4) |
| F4 (q odd) | q6(q2-1) | 5832 (q=3) | ≤7 | ≤1694 (q=3) |
| E6ε | q9(q2-1) | 1536 (q=2) | ≤9 | ≤1143 (q=2) |
| E7 | q15(q2-1) | 98304 (q=2) | ≤10 | ≤2550 (q=2) |
| E8 | q27(q2-1) | 402653184 (q=2) | ≤11 | ≤5621 (q=2) |
This proves that these groups have no non-trivial minimally active modules, but slightly better bounds are needed for the other groups, as the minimal faithful degrees are closer to qℓ.
Proposition 38.
Let G0 be the simple group G2(qt) for qt≥5, and let u be a p-element of G such that G=〈G0,u〉. There are no non-trivial, minimally active modules for G.
Proof.
Let G0 be a central extension of G2(qt) for some prime power q and some t≥1, and let G be obtained by adding on a field automorphism of order t to G0. Since G2(2)′=PSU3(3) we have already dealt with it, and we dealt with G2(3) and G2(4) in Proposition 36, we may assume that qt≥5.
First suppose that p≥5, so that t=1 or t≥5. From [10] we have α(G)≤5, and the Landazuri–Seitz bound [21] for G is qt(q2t-1). As p≥5, the order of u is at most one of tΦd(q) for d=1,2,3,6, with d=3 maximizing this, so we get
5t2(q2+q+1)>α(G)⋅(o(u)-1)≥dim(M)≥qt(q2t-1),
with t=1 and q=5 as solutions, and no solutions for t≥5. For G=G0=G2(5), the primes other than 2,3,5 dividing |G| are 7 and 31, each dividing it exactly once, whence we need a simple module of dimension at most 4⋅(31-1)=120, but 124 is the minimal degree.
Suppose that p=3, or p=2 and q is not a power of 3. The toral contribution is at most t(q+1), the Weyl contribution is at most 3, and the outer contribution is t. Since α(u)≤5, we get
15t2(q+1)>α(G)⋅(o(u)-1)≥dim(M)≥qt(q2t-1),
which has no solutions for qt≥5.
If p=2 and q is a power of 3, then we get the toral contribution to be t(q+1), the Weyl contribution to be 2, and the outer contribution to be 2t, so similar to the above expression, and we get
20t2(q+1)>qt(q2t-1),
which has no solutions for qt≥9. This completes the proof.
∎
Proposition 39.
Let G0 be the simple group D43(qt) for some q and t, and let u be a p-element of G such that G=〈G0,u〉. There are no non-trivial, minimally active modules for G.
Proof.
Let G0=D43(qt) for some prime power q and some t≥1 (there are no central extensions), and let G be obtained by adding on a field automorphism of order dividing 3t to G0 (see [7, Theorem 2.5.12]). Note that α(G)≤7 by [10, Proposition 5.7].
If p≥5, then the Sylow p-subgroup of G0 is abelian, so let d be the order of qt modulo p, so that d=1,2,3,6,12. If p∣Φ12(qt), then from the list of maximal subgroups in [18], we see that CG0(u) is abelian, so dim(M)≤2o(u) if M is minimally active, by Lemma 8. Furthermore, o(u)≤t2Φ12(q)=t2(q4-q2+1), and since dim(M)≥q3t(q2t-1) from [21], we get
2t2(q4-q2-1)≥dim(M)≥q3t(q2t-1),
which has only the solution qt=2, where p=13. Here dim(M)≥26=2o(u) by [16, p. 253] so there are no non-trivial minimally active modules here.
If d=1,2,3,6, then the toral contribution is at most t(q2+q+1), and the outer contribution is t; since α(u)≤7, we get that if M is minimally active, then
7t2(q2+q+1)≥q3t(q2t-1),
where qt=2 is again the only solution, this time with p=7. A quick computer check shows that for p=7 we actually have α(u)=2, so that there are no non-trivial minimally active modules here either.
Thus p=2,3 remain. If p=3, then the order of u is at most 9t2(q+1) by Proposition 26, using the fact that the exponent of the Weyl group of type D4 is 12. Since α(u)≤7, we have that
7⋅27t2(q+1)≥dim(M)≥q3t(q2t-1),
which is satisfied only for t=1, q=2,3, with q=3 not allowed as p∤q. If p=2, then u has order at most 4t2(q+1), and again α(u)≤7 so that
7⋅4t2(q+1)≥dim(M)≥q3t(q2t-1),
which only has a solution for t=1 and q=2, not of interest as p=2. Thus we need to consider p=3, G0=D43(2).
It is easy to check by computer that for any 3-element in G0, two conjugates of it generate G0, and o(u)≤9, so if u∈G0, then dim(M)≤2⋅8=16, smaller than the minimal dimension of 25 [16, p. 251]. If G/G0 has order 3, then the Sylow 3-subgroup still has exponent 9, so either o(u)=9, in which case two conjugates of u generate G and dim(M)≤16, or o(u)=3 and 〈u〉 lies outside G0, and then α(G)(o(u)-1)≤14, less than 25. This completes the proof.
∎
Proposition 40.
Let G0 be a central extension of a Ree or Suzuki group other than B22(8). There are no non-trivial, minimally active modules for smashG.
Proof.
Let G0 be a central extension of a Suzuki group B22(22n+1) for some n≥2, so that p is odd. The minimal faithful degree for G0 is 2n(22n+1-1) from [21], and from [10, Proposition 5.8] we have that α(G)≤3. Note also that there is no Weyl contribution as p is odd: the toral contribution is a divisor of one of 22n+1-1, 22n+1+2n+1+1 and 22n+1-2n+1+1, hence at most 22n+1+2n+1+1, and the outer contribution is t∣(2n+1).
This yields
3t(22n+1+2n+1+1)>α(u)⋅(o(u)-1)≥dim(M)≥2n(22n+1-1),
which has no solutions for t=1, and for t=2n+1 only works for n≤5. If t>1, then p∣(2n+1): if p∣(22n+1±2n+1+1) and p∣(2n+1), then p∣(2±2+1), so p=5. The other alternative is that p∣(22n+1-1), in which case p=1, which is not allowed. Thus p=5 always, so we need to consider n=2 only, as this is the only case where 5 divides 2n+1.
Here we just need to be more precise, noting that the Sylow 5-subgroup of B22(32).5 has order 125 but exponent 25, so actually dim(M)≤72, less than the minimal degree of 124.
We perform a similar analysis for the Ree groups G0=G22(32n+1) for n≥1, where the Landazuri–Seitz bound is 32n+1(32n+1-1), and from [10, Proposition 5.8] we have that α(G)≤3. If p=2, then o(u)=2 by Proposition 26, so dim(M)≤3 if M is minimally active, absurd; thus p is odd.
The order of any semisimple element of G0 is a divisor of one of 32n+1-1, 32n+1+1, 32n+1+3n+1+1 and 32n+1-3n+1+1, and the outer contribution is at most 2n+1: whence for u∈G,
3⋅(2n+1)⋅(32n+1+3n+1+1)≥3⋅(o(u)-1)≥dim(M)≥32n+1(32n+1-1),
if u acts minimally actively, which fails for all n≥1.
We end with the group G0=F42(22n+1). Here the Landazuri–Seitz bound is 29n+4(22n+1-1), and α(G)≤7. The toral contribution is at most one of
22n+1±1, 22n+1±2n+1+1, 24n+2±23n+2+22n+1±2n+1+1,
the Weyl contribution is 3, and the outer contribution divides 2n+1. Thus from the formula dim(M)≤α(G)(o(u)-1) for minimally active M, we get
21(2n+1)(24n+2+23n+2+22n+1+2n+1+1)>α(G)(o(u)-1)≥dim(M)≥29n+4(22n+1-1).
The only solution to this is n=0, i.e., G0 is the Tits group. Here it is easy to check that G0 is generated by two conjugates of any element of order at least 3, that u has order at most 13, and that dim(M)≥26, thus there is no example here.
∎
