1 Introduction
Throughout this paper all groups are assumed to be finite.
The minimal faithful permutation degreeμ(G) of a group G is the smallest nonnegative integer n such that G embeds in the symmetric group Sym(n). Note that μ(G)=0 if and only if G is trivial. It is well known (and referred to as Karpilovsky’s theorem, see, for example, [11, 12]) that if G is a nontrivial abelian group, then μ(G) is the sum of the prime powers that occur in a direct product decomposition of G into cyclic factors of prime power order. Johnson proved (see [11, Theorem 1]) that the Cayley representation of a group G is minimal, that is, μ(G)=|G|, if and only if G is cyclic of prime power order, the Klein four-group or a generalised quaternion 2-group. A number of other explicit calculations of minimal degrees and a variety of techniques appear in Johnson [11], Wright [21, 22], Neumann [15], Easdown and Praeger [3], Kovacs and Praeger [13], Easdown [2], Babai, Goodman and Pyber [1], Holt [9], Holt and Walton [10], Lemieux [14], Elias, Silbermann and Takloo-Bighash [5], Franchi [6], Saunders [18, 17, 19, 20] and Easdown and Saunders [4]. This present article, building on work initiated by the second author in [8], focuses on minimal degrees of semidirect products of groups, proves a reduction theorem (see Theorem 2.7 below) and provides exact formulae (see Theorems 4.5 and 4.8 below) for minimal degrees in the case when the base group is an elementary abelian p-group and the extending group is cyclic of order q where p and q are different primes.
For any groups G and H and subgroups S of G, we always have the inequalities
(1.1)μ(S)≤μ(G)
and
(1.2)μ(G×H)≤μ(G)+μ(H).
Many sufficient conditions are known for equality to occur in (1.2), for example, when G and H have coprime order (Johnson [11, Theorem 1]), when G and H are nilpotent (Wright [22]), when G and H are direct products of simple groups (Easdown and Praeger [3]), and when G×H embeds in Sym(9) (Easdown and Saunders [4]).
The first published example where the inequality in (1.2) is strict appears in Wright [22], where G×H is a subgroup of Sym(15).
Saunders [17, 18] describes an infinite class of examples, which includes the example in [22] as a special case, where strict inequality takes place in (1.2). The smallest example in his class occurs when G×H embeds in Sym(10). In all of these examples of strict inequality, the groups G and H have the properties that H is cyclic of prime order and
(1.3)μ(G×H)=μ(G).
As an application of our three main theorems, the article culminates (see Example 5.8 below) in an infinite class of examples where (1.3) occurs, where H may be a product of elementary abelian groups with an arbitrarily large number of factors and different prime exponents and G does not decompose as a nontrivial direct product.
Recall that if G is nontrivial, then μ(G) is the
smallest sum of indices for a collection of subgroups 𝒞={H1,…,Hk} such that ⋂i=1kHi is core-free. In this case we say that 𝒞affords a minimal faithful representation ofG.
The subgroups H1,…,Hk become the respective point-stabilisers for the action of G on its orbits and letters in the ith orbit may be identified with cosets of Hi for i=1…,k.
If k=1, then the representation afforded by 𝒞 is transitive and H1 is a core-free subgroup.
Remark 1.1It follows quickly that if G is a group with unique subgroups of orders p1,…,pk respectively, where p1,…,pk are distinct primes, then μ(G)≥|G|p1+…+|G|pk, where |G|p denotes the largest power of p dividing |G|. For example, suppose G is the generalised quaternion group of order 4n for n≥2, given by the presentation
G=Q4n=〈a,b∣a2n=b4=1,an=b2,ab=a-1〉.
Then 〈b2〉 is the unique subgroup of G of order 2.
If n is a power of 2, then μ(G)≥|G|2=|G|, whence μ(G)=|G|, the only nonabelian case where this is possible (see Johnson [11, Theorem 2]). Suppose then that n is not a power of 2 and
let p1,…,pk be the odd prime divisors of n. Then 〈a2n/pi〉 is the unique subgroup of G of order pi for i=1,…,k, so
μ(G)≥|G|2+|G|p1+…+|G|pk.
Write
|G|=2mp1α1…pkαk,
where m≥2 and α1,…,αk≥1, and put
H=〈a2m-1〉 and Hi=〈apiαi,b〉 for i=1,…,k.
Then
{H,H1,…,Hk} affords a faithful representation of G so that
μ(G)=|G|2+|G|p1+…+|G|pk.
Note that if m=2, then |a2|=n is odd, G=〈a2,b〉 and the presentation above simplifies, replacing a2 by x:
G=〈x,b∣xn=b4=1,xb=x-1〉,
so that G becomes a semidirect product. If we put n=3, then μ(G)=3+4=7 and G becomes the smallest group with the property that it does not have a nilpotent subgroup with the same minimal degree. The class of groups that do have nilpotent subgroups with the same degree was introduced by Wright [22], and its pervasiveness within the class of permutation groups of small degree was an important tool in [4].
2 Preliminaries on semidirect products
Recall that a group G is an internal semidirect product of a normal subgroup N by a subgroup H if G=NH and N∩H is trivial, in which case the conjugation action of N on H induces a homomorphism φ:N→Aut(H). Conversely, if N and H are any groups and φ:H→Aut(N) any homomorphism, then the cartesian product of sets
N⋊H=N⋊φH={(n,h)∣n∈N,h∈H}
becomes a group, called the external semidirect product, under the binary operation
(n1,h1)(n2,h2)=(n1(n2(h-1φ)),h1h2),
in which case N⋊H becomes an internal semidirect product of a copy of N by a copy of H and we may write N⋊H=NH without causing confusion.
Remark 2.1It is well known that G⋊φH embeds in Sym(G)×H, and in Sym(G) if φ is injective. Hence μ(G⋊H) is bounded by |G|+μ(H) always, and by |G| if φ is injective (though see Lemma 2.2 below for an alternative proof).
The bound |G|+μ(H) can easily be achieved, for example, whenever the semidirect product is direct (that is, φ is trivial), G any group for which the Cayley representation is minimal and H any group of order coprime to |G|. For a class of semidirect products that are not direct, let G=Cpn and H=Cq2, where p and q are distinct primes and n a positive integer such that q>pn-1. Put H=〈c〉 and suppose we have a homomorphism φ:H→Aut(G) such that |cφ|=q, so φ is neither trivial nor injective, and that the conjugation action induced on G is irreducible. A simple subclass of examples would be when (p,q,n)=(p,2,1) and cφ the inversion automorphism of G (so of order 2). (An instance of this, when (p,q,n)=(3,2,1), features in Example 2.8 below.) It follows, by observations in Remark 1.1, that
μ(G⋊H)=|G|+μ(H)=pn+q2.
For example, if (p,q,n)=(5,2,1), then μ(S)=5+4=9 and we get the intransitive representation
S≅C5⋊C4≅〈(12345),(15)(24)(6789)〉.
Lemma 2.2Let K be an internal semidirect product of G by H. Then core(H) equals kerφ, where φ:H→Aut(G) is the homomorphism induced by conjugation. In particular, if φ is injective, then H is core-free and {H} affords a transitive representation of K of degree |G|, so that μ(K)≤|G|.
Proof.
Certainly kerφ is a normal subgroup of K contained in H, so we have that kerφ≤core(H). Conversely, elements of core(H) commute with elements of G, so core(H)≤kerφ.
∎
It will be useful, in verifying the first alternative of the main formula (4.1) below, to note that, under certain conditions, the minimal degree of the semidirect product coincides with the minimal degree of the base group:
Lemma 2.3Suppose that G⋊φH is a semidirect product of groups such φ is injective.
If G has a minimal faithful representation afforded by a collection of subgroups that are invariant under the conjugation action of H, then
μ(G⋊H)=μ(G).
Proof.
We may regard G⋊H=GH as an internal semidirect product. Since φ is injective, H is core-free by Lemma 2.2. Suppose that {B1,…,Bk} is a collection of subgroups of G invariant under conjugation by H and affords a minimal faithful representation of G.
For i=1 to k, put Di=BiH, which is a subgroup of GH of index |G:Bi|. Then
{D1,…,Dk} affords a faithful representation of GH of degree
|G:B1|+…+|G:Bk|=μ(G).
But μ(GH)≥μ(G), so we have equality.
∎
Example 2.4Let p and q be primes such that the field 𝔽p={0,…,p-1} has a primitive qth root ζ of 1. Let φ:Cq→Aut(Cpq) be the homomorphism induced by the map
cφ:(x1,x2,…,xq)↦(x1,x2ζ,x3ζ2,…,xqζq-1),
where c is a generator of Cq and x1,…,xq∈Cp. Put G=Cpq⋊φCq.
We may write G=KC as an internal semidirect product of K≅Cpq by C≅Cq, where K is an internal direct product H1…Hq, where Hi≅Cp for i=1,…,q. Put
Hi^=H1…Hi-1Hi+1…Hq,
which is a subgroup of K of index p, for i=1,…,q. Put 𝒞={H1^,…,Hq^}. Then ∩𝒞 is trivial, so 𝒞 affords a faithful representation of K of degree pq.
But each Hi^ is invariant under the conjugation action by C, so μ(G)=pq, by Lemma 2.3.
It is interesting that in this case we can also find a faithful transitive representation of G by letting ai be a generator for Hi for each i and putting
H={a1i1…aqiq∈H1…Hq∣i1+…+iq=0}.
Then H is a core-free subgroup of G (in fact, a canonical codimension 1 subspace of the additive vector space corresponding to the base group, in the sense of Lemma 3.6 below) of index pq.
Consider groups H and K of coprime order and C a cyclic group such that |C| and |H||K| are also coprime. Let φ:C→Aut(H×K) be a homomorphism, so that we may form the semidirect product
G=(H×K)⋊C=(H×K)⋊φC.
Let φH:C→Aut(H) and φK:C→Aut(K) where, for all h∈H, k∈K and c∈C,
(2.1)(h,k)(cφ)=(h(cφH),k(cφK)),
so that we have the related semidirect products
H⋊C=H⋊φHC and K⋊C=K⋊φKC.
If φ is trivial, then G≅H×K×C. If φH is trivial, then G≅H×(K⋊C). If φK is trivial, then G≅(H⋊C)×K. Note that always G embeds in the direct product (H⋊C)×(K⋊C) under the map
((h,k),c)↦((h,c),(k,c))
for all h∈H, k∈K, c∈C, so that, by (1.1) and (1.2),
(2.2)μ(G)≤μ((H⋊C)×(K⋊C))≤μ(H⋊C)+μ(K⋊C).
In Theorem 2.7 below, we show that equality occurs throughout (2.2) when both φH and φK are nontrivial and C≅Cq for some prime q. We first establish some useful general facts.
Lemma 2.5Let G=HC be an internal semidirect product of a normal subgroup H by a cyclic subgroup C≅Cq for some prime q not dividing |H|. Let K be a subgroup of G that is not a subgroup of H.
There exists g∈G such that K=(H∩K)Cg is an internal semidirect product of H∩K by Cg.
If H∩K is normal in H, then H∩K is normal in G.
If K is normal in G, then K=(H∩K)C.
Proof.
Part (a) follows by Sylow’s theorem, and then parts (b) and (c) are immediate.
∎
Lemma 2.6Let G=HC be an internal semidirect product that is not direct of a normal subgroup H by a cyclic subgroup C≅Cq for some prime q not dividing |H|. Then any collection C affording a minimal faithful representation of G does not contain any normal subgroup of G that is a subgroup of H.
Proof.
Let 𝒞={K1,…,Kk} afford a minimal faithful representation of G. Suppose, by way of contradiction, that 𝒞 contains a subgroup of H that is normal in G. Without loss of generality, we may assume that K1≤H and K1 is normal in G.
If K1≠H, then {K1C,H,K2,…,Kk} affords a faithful representation of degree smaller than that afforded by 𝒞, contradicting minimality. Hence K1=H. If k=1, then H=core(H)={1},
which is impossible. Hence k>1. Put N=core(K2∩…∩Kk), so H∩N={1}. If q does not divide |N|, then N≤H, so N={1} and {K2,…,Kk} affords a faithful representation, again contradicting minimality. Hence q divides |N|, so, by Lemma 2.5 (c), we have N=(H∩N)C=C, yielding a final contradiction, since C is not normal.
∎
The following theorem reduces calculations of minimal degrees of semidirect products by a q-cycle, where q is a prime that does not divide the order of the base group, to those cases where the base group is a p-group for p≠q.
Theorem 2.7Let G=(H×K)⋊C be a semidirect product where H and K are groups of coprime order and C≅Cq for some prime q not dividing |H||K|. Then
μ(G)={μ(H)+μ(K)+qif φ is trivial,μ(H)+μ(K⋊C)if φH is trivial,μ(H⋊C)+μ(K)if φK is trivial,μ(H⋊C)+μ(K⋊C)if neither φH nor φK is trivial.
Proof.
Note that the first case is a special case of the second and third cases, and the formulae for the first three cases follow by Johnson’s result [11, Theorem 1] that μ is additive with respect to taking direct products of groups of coprime order.
Suppose then that neither φH not φK are trivial. We may regard G=HKC as an internal semidirect product of HK by C, where HK is an internal direct product of H and K. By (2.2), it suffices to prove
(2.3)μ(G)≥μ(HC)+μ(KC).
Let 𝒞 be a collection of subgroups of G that affords a minimal faithful permutation representation of G.
Since |H| and |K| are coprime, subgroups of HK have the form H0K0 for some H0≤H and K0≤K. By Lemma 2.5 (a), subgroups of G that are not subgroups of HK have the form H0K0Cg for some H0≤H, K0≤K and g∈G, such that H0K0 is normal in H0K0Cg.
By a result of Johnson [11, Lemma 1], we may assume that each element of 𝒞 is meet-irreducible, that is, does not decompose as the intersection of two larger subgroups. Therefore, elements of 𝒞 have the form
H0K,HK0,H1KCx or HK1Cy
for some H0,H1≤H, K0,K1≤K and x,y∈G. In these respective cases, note that
coreG(H0K)=coreHC(H0)K,coreG(HK0)=HcoreKC(K0),
and, by Lemma 2.5 (c),
coreG(H1KCx)={coreHC(H1)KCif q divides |coreG(H1KCx)|,coreHC(H1)Kotherwise,
and
coreG(HK1Cy)={HcoreKC(K1)Cif q divides |coreG(HK1Cy)|,HcoreKC(K1)otherwise.
Put
𝒟H={H0∣H0≤HandH0K∈𝒞},
ℰH={H1C∣H1≤HandH1KCx∈𝒞forsomex∈G},
𝒟K={K0∣K0≤KandHK0∈𝒞},
ℰK={K1C∣K1≤KandHK1Cy∈𝒞forsomey∈G}.
By inspection, the index sum of elements of 𝒞 in G is equal to the index sum of elements of 𝒟H∪ℰH in HC added to the index sum of elements of 𝒟K∪ℰK in KC. Hence, to complete the proof of (2.3), it suffices to show that 𝒟H∪ℰH and 𝒟K∪ℰK afford faithful representations of HC and KC respectively.
Observe that
coreHC(⋂H0∈𝒟HH0∩⋂H1C∈ℰHH1)K∩HcoreKC(⋂K0∈𝒟KK0∩⋂K1C∈ℰKK1)
⊆coreG(⋂𝒞)={1}.
In particular,
coreHC(⋂H0∈𝒟HH0∩⋂H1C∈ℰHH1)={1}.
If 𝒟H≠∅ then, since ⋂𝒟H⊆H, we have
coreHC(⋂(𝒟H∪ℰH))⊆coreHC(⋂H0∈𝒟HH0∩⋂H1C∈ℰHH1)={1}.
Suppose that 𝒟H=∅. If ℰH=∅, then 𝒟K∪ℰK≠∅ so that
H⊆coreG(⋂𝒞)={1},
which is impossible. Hence ℰH≠∅ and
coreHC(⋂H1C∈ℰHH1)={1}.
If coreHC(H1C) contains an element of order q for all H1C∈ℰH then, in each case,
coreHC(H1C)=coreHC(H1)C, so that
C=coreHC(⋂H1C∈ℰHH1)C=⋂H1C∈ℰHcoreHC(H1C)
is a normal subgroup of HC, contradicting that φH is nontrivial. Hence, for at least one H1C∈ℰH, we have coreHC(H1C)=coreHC(H1), so that
coreHC(⋂ℰH)=coreHC(⋂H1C∈ℰHH1C)
=coreHC(⋂H1C∈ℰHH1)={1}.
This proves that 𝒟H∪ℰH affords a faithful representation of HC. Similarly
𝒟K∪ℰK affords a faithful representation of KC, and this completes the proof of (2.3).
∎
Example 2.8Let G be the holomorph of C3×C5, that is,
G=(C3×C5)⋊idAut(C3×C5)≅(C3×C5)⋊(C2×C4).
We may regard G=HKCD as an internal semidirect product of a direct product HK by another direct product CD, where H=〈h〉≅C3, K=〈k〉≅C5, C=〈c〉≅C2≅Aut(C3) and D=〈d〉≅C4≅Aut(C5). Then
μ(G)≥μ(C3×C5)=8
and
G≅〈(123),(45678),(12),(4576)〉,
which verifies that μ(G)=8. Put C1=〈cd2〉, C2=〈cd〉, G1=HKC1 and G2=HKC2. Then
G1≅(C3×C5)⋊φC2≅〈(123),(45678),(12)(47)(56)〉,
where φ induces conjugation action that is inversion, and both C3⋊φ1C2 and C5⋊φ2C2 are dihedral, where φ1=φC3 and φ2=φC5 are defined by (2.1), and both nontrivial. As predicted by Theorem 2.7,
μ(G1)=8=3+5=μ(C3⋊φ1C2)+μ(C5⋊φ2C2).
However,
G2≅(C3×C5)⋊ψC4≅〈(123),(45678),(12)(4576)〉,
where C3⋊ψ1C4 is generalised quaternion of degree 7 (see Remark 1.1) and C5⋊ψ2C4 has degree μ(C5)=5, by Lemma 2.3, where ψ1=ψC3 and ψ2=ψC5 are defined by (2.1). Here
μ(G2)=8<12=7+5=μ(C3⋊ψ1C4)+μ(C5⋊ψ2C4).
This is the smallest example where we do not get equality throughout in (2.2), yet all of the homomorphisms defining the semidirect products are nontrivial.
3 Preliminaries on group actions on a vector space
The aim in this section is to develop machinery to calculate, in the next section, minimal degrees of all semidirect products of elementary abelian p-groups by cyclic groups of order q where p and q are different primes, exploiting the fact that
group actions may be analysed using standard methods from linear algebra.
Let V be an n-dimensional vector space over 𝔽p, written additively, and T:V→V an invertible linear transformation.
Define the semidirect product of V by 〈T〉
(or more simply the semidirect product of V by T)
to be
(3.1)V⋊T=V⋊〈T〉={(v,Ti)∣v∈V,i∈ℤ},
with binary operation
(3.2)(v,Ti)(w,Tj)=(v+Ti(w),Ti+j),
for v,w∈V and i∈ℤ. Then V⋊T becomes a group.
A subspace of V that is T-invariant is referred to simply as invariant. Thus invariant subspaces of V become normal subgroups of V⋊T.
We define the core of any subspace W of V, denoted by core(W), to be the largest invariant subspace of V contained in W.
Thus core(W)=coreG(W), in the usual sense, that is, the largest normal subgroup of G contained in W, where G=V⋊T.
We suppose throughout, unless stated otherwise, that T≠id and Tq=id, where id is the identity linear transformation and q is a prime different to p.
The characteristic and minimal polynomials of T are referred to as χT=χT(x) and φT=φT(x) respectively.
By choosing a basis for V we may identify V with the vector space 𝔽pn of column vectors of length n with entries from 𝔽p and T with the n×n matrix of the linear transformation with respect to the basis, and so regard T(v)=Tv as a matrix product. Under these identifications
V⋊T≅Cpn⋊φCq
under the map
([λ1⋮λn],Ti)↦((aλ1,…,aλn),b-i),
where we write Cp=〈a〉, Cq=〈b〉, and φ:Cq→Aut(Cpn) is the homomorphism induced by
bφ:(aλ1,…,aλn)↦(aλ1′,…,aλn′),
where
T[λ1⋮λn]=[λ1′⋮λn′].
Lemma 3.1Let T1 and T2 be n×n matrices over Fp of multiplicative order q and put V=Fpn for some positive integer n. Then V⋊T1≅V⋊T2 if and only if T1 and some power of T2 are similar. In particular, if T1 and T2 are similar, then V⋊T1≅V⋊T2.
Proof.
If T1 and T2k are similar, for some k∈ℤ, then k≠0 modulo q,
T1 equals P-1T2kP
for some invertible matrix P, and the mapping
(v,T1i)↦(Pv,T2ki),
for v∈V and i∈ℤ, is an isomorphism.
Conversely, if θ:V⋊T1→V⋊T2 is an isomorphism, then
(0,T1)θ=(w,T2k)
for some w∈V and integer k, and one may check that
T1 and T2k are similar.
∎
Thus, in calculating minimal degrees later, we may assume T is in primary rational canonical form. By Maschke’s theorem, since p does not divide q=|〈T〉|, all invariant subspaces of V have invariant complements, so that the minimal polynomial φT is square-free with regard to irreducible factors. All blocks in the primary rational canonical form of T become companion matrices of monic irreducible polynomials, and the restriction of T to an indecomposable subspace of V will always have an irreducible minimal polynomial. The canonical form is thus characterised uniquely, up to the order of blocks, by χT.
The number of blocks corresponding to one particular irreducible factor is just the multiplicity of that factor in χT. An irreducible factor of φT=φT(x) divides xq-1, so is either x-1 or a polynomial of the form
(3.3)πα(x)=(x-α)(x-αp)…(x-αps-1),
where s is the multiplicative order of p modulo q and α is a primitive qth root of 1 in an extension field 𝔽=𝔽p(α) of 𝔽p (where 𝔽=𝔽p if s=1).
Remark 3.2The previous lemma in principle allows for nontrivial determination of isomorphism between semidirect products in our class. For example, take n=6, p=13 and q=7, so that s=2. Consider the following irreducible polynomials over 𝔽13:
r1=x2+3x+1,
r2=x2+6x+1,
r3=x2+5x+1.
Put π1=r12r2, π2=r22r3 and π3=r12r3. Let Ti be the companion matrix for πi and Gi=𝔽136⋊Ti, for i=1,2,3. There exists a primitive 7th root α in an extension 𝔽 of 𝔽13 such that
r1=(x-α)(x-α6),
r2=(x-α2)(x-α5),
r3=r3(x)=(x-α3)(x-α4).
It follows that T2 and T12 are similar, but T3 is not similar to any power of T1. Hence, G1≅G2, but G1≇G3, by Lemma 3.1.
The following two lemmas are probably well known.
Lemma 3.3Let W be a subspace of a vector space V. Suppose V=K⊕K′ for some subspaces K and K′ such that K is also a subspace of W. Put L=W∩K′. Then
W=K⊕L.
The codimension of L in K′ is the same as the codimension of W in V. If, further, T:V→V is a linear transformation and K is the core of W with respect to T, then L is core-free.
Proof.
All of the claims follow quickly from the definitions.
∎
Lemma 3.4Let T:V→V be an invertible linear transformation such that φT has degree d. Let W be a subspace of V of codimension k. Then core(W) has codimension at most kd. In particular, if W has codimension 1, then core(W) has codimension at most d.
Proof.
The claim follows from the fact that
core(W)=W∩T(W)∩…∩Td-1(W)
and W,T(W),…,Td-1(W) all have the same codimension in V, since T is invertible.
∎
Proposition 3.5Let T:V→V be an invertible linear transformation of a finite-dimensional vector space V such that φT is a product of distinct irreducible factors. Let
W be a codimension 1 subspace of V. Then any invariant complement of core(W) in V is a sum of indecomposable subspaces with distinct minimal polynomials.
Proof.
Let φT(x)=r1(x)…rm(x), where r1,…,rm are the distinct irreducible factors. Put Vi=ker(ri(T)) and Wi=core(W)∩Vi for i=1,…,m. Then we have core(W)=W1⊕…⊕Wm. Let i∈{1,…,m}. Let ki be the number of indecomposable components of V having minimal polynomial ri, which is just the number of indecomposable components of Vi. To complete the proof, therefore, by the Krull–Schmidt theorem, it suffices to show that the number of indecomposable components of Wi is ki or ki-1. Let di be the degree of ri. Observe that Wi=coreVi(W∩Vi). But W∩Vi has codimension at most 1 in Vi. Thus Wi has codimension at most di in Vi, by Lemma 3.4. But di is the dimension of any indecomposable component of Vi, so Wi contains at least ki-1 indecomposable components.
∎
Lemma 3.6Let T:V→V be a linear transformation such that
φT=r1…rm for distinct irreducible polynomials r1,…,rm.
Suppose that V=V1⊕…⊕Vm, where Vi=ker(ri(T)) is indecomposable for i=1,…,m. Let Bi be a basis for Vi for i=1,…,m and put B=B1∪…∪Bm, which is a basis for V. Put
V¯={∑b∈Bλbb∈V|∑b∈Bλb=0}.
Then V¯ is a core-free subspace of codimension 1.
Conversely, if W is a core-free subspace of codimension 1, then we can choose a basis Bi for Vi for i=1,…,m such that W=V¯.
Proof.
Put n=dim(V). If n=1, then the claims hold trivially, so we may suppose n≥2.
If B={v1,…,vn}, then {v1-v2,…,v1-vn} is a basis for V¯, so dim(V¯)=n-1. Because r1…,rm are distinct, V1,…,Vm are the unique indecomposable subspaces, and none of these is contained in V¯, so core(V¯)={0}.
Conversely, let W be a codimension 1 subspace of V such that core(W)={0}. Choose any basis B1′ for W∩V1. Certainly, W∩V1 has codimension 1 in V1, since core(W)={0}. Hence B1′∪{v1} is a basis for V1 for some v1∈V1. Put
B1={b+v1∣b∈B1′}∪{v1}.
Then B1 is also a basis for V1. If m=1, then V=V1 and V¯=W, starting an induction. Suppose m>1 and
put V^=V2⊕…⊕Vm, so that V=V1⊕V^. Certainly, W∩V^ has codimension 1 in V^, since core(W)={0}.
Suppose, as an inductive hypothesis, that we have bases B2,…,Bm for V2,…,Vm respectively, such that
W∩V^={∑c∈Cλcc∈V^|∑c∈Cλc=0},
where C=B2∪…∪Bm. Observe that (W∩V1)⊕(W∩V^) has codimension 1 in W, so we may choose some
w∈W\((W∩V1)⊕(W∩V^)).
But w=v+v^ for some unique w∈V1 and v^∈V^. If one of v or v^ is in W, then both are, contradicting the choice of w. Hence v,v^∉W. But v^=∑c∈Cλcc for some scalars λc. Put
λ=∑c∈Cλc.
By the inductive hypothesis, λ≠0. Now put
B1={b-1λv|b∈B1′}∪{-1λv},
so that B1 is a basis for V1.
Finally, put B=B1∪…∪Bm and form V^ with respect to B.
But,
w=v+v^=-λ(-1λv)+∑c∈Cλcc
and
-λ+∑c∈Cλc=-λ+λ=0,
so that w∈V¯, by definition.
Noting that
W=〈w〉⊕(W∩V1)⊕(W∩V^),
it is straightforward, using the inductive hypothesis, to verify that W⊆V¯. Because dim(W)=n-1=dim(V¯), we have W=V¯, establishing the inductive step.
∎
We call the subspace V¯ defined in the statement of the previous lemma, the canonical core-free subspace associated with V (depending of course on the choice of basis).
Proposition 3.7Let W be a subspace of a finite-dimensional vector space V over Fp acted on by an invertible linear transformation T:V→V of order q, where p and q are distinct primes. Then W has codimension 1 if and only if some (and hence every) invariant complement core(W)′ of core(W) in V is a sum of indecomposable components with distinct minimal polynomials such that
W=core(W)⊕core(W)′¯
for some canonical core-free subspace core(W)′¯ of core(W)′.
Proof.
Note first that the hypotheses guarantee that T is invertible and φT is a product of distinct irreducible polynomials. The “if” direction is immediate by Lemma 3.6. Suppose then that W has codimension 1, and choose some invariant complement core(W)′ of core(W) in V.
By Proposition 3.5, the indecomposable components of core(W)′ have distinct minimal polynomials. By Lemma 3.3,
W=core(W)⊕(W∩core(W)′),
and
W∩core(W)′ is core-free of codimension 1 in core(W)′. By Lemma 3.6, there is a choice of basis for core(W)′ such that W∩core(W)′=core(W)′¯.
∎
4 Minimal degrees when the base group is elementary abelian
Throughout this section p and q are distinct primes. Let V=𝔽pn≅Cpn be an n-dimensional vector space over the field 𝔽p of p elements, for some fixed positive integer n, and T an n×n matrix with entries from 𝔽p of multiplicative order q.
Recall that, if W is a subspace of V that is invariant under this action, then W has an invariant complement W′ in V.
The minimal polynomial φT
is a product of distinct irreducible polynomials, all of degree s where s is the multiplicative order of p modulo q, with the possible exception (when s≥2) of a factor x-1. Note that s=1 if and only if 𝔽p has a primitive qth root of unity, in which case all the irreducible factors of φT are linear.
Proposition 4.1Let G=V⋊T.
There exist nonnegative integers ℓ and t and a collection C=D∪E affording a minimal faithful representation of G such that
𝒟={D1,…,Dℓ} 𝑎𝑛𝑑 ℰ={E1〈T〉,…,Et〈T〉}
for some codimension 1 subspaces D1,…,Dℓ of V, and invariant subspaces E1,…,Et of V, such that each of E1,…,Et complements an indecomposable subspace (where we interpret ℓ=0 and t=0 to mean D=∅ and E=∅ respectively).
Note that it is possible to have t=1 and E1={0}, the complement of V in the case that V is indecomposable.
Proof.
We may regard G=VC as an internal semidirect product of V by C≅〈T〉≅Cq, but still retaining vector space terminology and additive notation for the group operation restricted to V.
By [11, Lemma 1] there exists a collection 𝒞 of meet-irreducible subgroups affording a minimal faithful representation of G. Then 𝒞=𝒟∪ℰ, where 𝒟, possibly empty, comprises all subgroups in 𝒞 of index divisible by q, and ℰ, possibly empty, consists of all subgroups in 𝒞 of order divisible by q.
In particular, elements of 𝒟 are subgroups of V. By Lemma 2.6, these must all be proper subgroups of V, since V is normal in G, so, being meet-irreducible, must have codimension 1 as subspaces of V.
Let K∈ℰ, so q divides |K|. Put W=K∩V.
Note that V is elementary abelian, so all of its subgroups are normal in V. By (a) and (b) of Lemma 2.5, K=W〈T〉g for some g∈G and W is an invariant subspace of V (being normal in G).
Certainly W≠V (for otherwise G=K∈𝒞, contradicting minimality), so V=W⊕W′ for some nontrivial invariant subspace W′ of V. If W′ is not indecomposable, then W′=W1′⊕W2′ for some nontrivial invariant subspaces W1′ and W2′ of V, so
W=(W⊕W1′)∩(W⊕W2′)
and K=K1∩K2, where K is a proper subgroup of Ki=(W⊕Wi′)〈Tg〉 for i=1 and 2, contradicting that K is meet-irreducible. Hence W′ is indecomposable. Note that K and W〈T〉 have the same core and index in G, so we may, if necessary, replace K by W〈T〉 in ℰ.
∎
In what follows we develop a complete catalogue, namely,
(4.1) and (4.8) below,
of formulae for μ(V⋊T).
Note, throughout, that T≠I, so φT(x)≠x-1.
The next two theorems cover all possibilities, where s is the order of p modulo q. In the first case (Theorem 4.5), we investigate what happens when all of the factors of the minimal polynomial have the same degree s≥1.
In the second case (Theorem 4.8), we investigate the remaining possibilities, namely, when x-1 is a factor and all other factors have the same degree s≥2.
Lemma 4.2If G=V⋊T, where all irreducible factors of φT are linear, then μ(G)=np.
Proof.
Suppose that all irreducible factors of φT are linear. Without loss of generality, we may suppose T is diagonal and V=〈v1,…,vn〉, where v1,…,vn are eigenvectors for T.
For i=1,…,n, put
Hi=〈v1,…,vi-1,vi+1,…,vn〉.
Then {H1,…,Hn} affords a minimal faithful representation of V by T-invariant subspaces of degree np. By Lemma 2.3, μ(G)=μ(V)=np.
∎
An illustration of the phenomenon of Lemma 4.2 appears above in Example 2.4.
Lemma 4.3Let p and q be distinct primes and s the multiplicative order of p modulo q. Suppose that s≥2. Let a be the smallest integer such that q<aps-1. Then a=1, or a=2 and q=1+p+…+ps-1. If s=a=2, then p=2 and q=3.
Proof.
Suppose a>1, so ps-1<q. Note that q divides ps-1=(p-1)(1+p+…+ps-1). If q divides p-1, then q<p≤ps-1, a contradiction. Hence q divides 1+p+…+ps-1 and ps-1<q<1+p+…+ps-1. It follows that q=1+p+…+ps-1<2ps-1 and a=2.
∎
Remark 4.4A generalised Mersenne primeq has the form q=1+p+…+pk-1 for some prime p and integer k (which includes the usual Mersenne primes of the form 2k-1). The previous lemma asserts that, in our context, if a=2 and s≥2, then q must be a generalised Mersenne prime. It is not known if there are infinitely many such primes.
Theorem 4.5Suppose that r1,…,rm are distinct irreducible polynomials over Fp of degree s, where s is the order of p modulo q, such that
φT=r1…rm 𝑎𝑛𝑑 χT=r1k1…rmkm.
We may suppose k1≥k2≥…≥km. Then
(4.1)μ(V⋊T)={npif s=1,k1pqif s>1 and q<ps-1,k1psif s>1,m=1 and q>ps-1,k2pq+(k1-k2)psif s>1,m>1 and q>ps-1.
Proof.
The first alternative in (4.1) is given by Lemma 4.2, so we may suppose s>1. Let a denote the smallest integer such that q<aps-1. By Lemma 4.3, a=1 or 2. It is convenient, throughout, to put km+1=0. In particular, if m=1 and a=2, then ka=k2=0.
Put G=V⋊T=V〈T〉 (regarded as an internal semidirect product, mixing addition and multiplication, without ever causing confusion).
We have a direct sum decomposition
V=⊕i=1m⊕i=1kiVij=⊕(i,j)∈IVij,
where Vij is an indecomposable subspace of V such that T|Vij has minimal polynomial ri for each (i,j)∈I, where
I={(i,j)∣1≤i≤m, 1≤j≤ki}.
For J⊆I, put
VJ=⊕(i,j)∈JVij,
so that V=VI=VJ⊕VI\J. If W=VJ for some J⊆I, then put W′=VI\J, so that V=W⊕W′.
Note that if ka=0, then m=1 and a=2. Suppose for the time being that ka≥1, so either a=1, or a=2 and m≥2.
Because ka≥ka+1≥…≥km>km+1=0, we have that, for each
j=1 to ka, there exists some largest ℓj∈{a,…,m} such that
kℓj≥j≥kℓj+1,
and we put
Wj=⊕i=1ℓjVij,
so that T|Wj has minimal polynomial r1…rℓj. In particular, we have ℓ1=m, since km≥1>0=km+1, and T|W1 has minimal polynomial r1…rm.
Thus
(4.2)V=VX⊕⊕j=1kaWj,
where X={(1,j)∣k2<j≤k1} if a=2 and k1>k2, and X=∅ otherwise, in which case we interpret VX={0}.
For j=1 to ka, put
Hj=Wj¯⊕Wj′,
where Wj¯ is a canonical codimension 1 subspace of Wj as described in Lemma 3.6, so that core(Wj¯)={0}, core(Hj)=Wj′ and |G:Hj|=pq.
For (1,j)∈X, put
Kj=V1j′〈T〉,
so that core(Kj)=V1j′ and |G:Kj|=ps. Now put
(4.3)𝒞={H1,…,Hka}∪{Kj∣(1,j)∈X}.
Then
core(⋂𝒞)=⋂j=1kaWj′∩⋂(1,j)∈XV1j′=VX∩VX′={0},
so that 𝒞 affords a faithful representation of G of degree
∑j=1ka|G:Hj|+∑(1,j)∈X|G:Kj|=kapq+(k1-ka)ps.
Note that if ka=0, so that m=1 and a=2, then (4.2) may be interpreted as V=VI (since X=I) and (4.3) may be interpreted as 𝒞={Kj∣(1,j)∈I}, and the conclusion about the faithfulness and degree of the representation afforded by 𝒞 still holds. This proves that, in all cases,
μ(G)≤kapq+(k1-ka)ps.
We now prove that this formula is also a lower bound for μ(G).
By Proposition 4.1, there exists a collection 𝒞=𝒟∪ℰ affording a minimal faithful representation of G such that
𝒟={D1,…,Dℓ} and ℰ={E1〈T〉,…,Et〈T〉}
for some codimension 1 subspaces D1,…,Dℓ of V, and invariant subspaces E1,…,Et of V, each of which complements an indecomposable subspace. We interpret ℓ=0 and t=0 to mean 𝒟=∅ and ℰ=∅ respectively.
By Proposition 3.7, for i=1,…,ℓ, we may write
Di=core(Di)⊕core(Di)′¯=Si¯⊕Si′,
where we put Si=core(Di)′.
The degree of the representation afforded by 𝒞 is ℓpq+tps, so to complete the proof of the theorem it suffices to show
(4.4)ℓpq+tps≥kapq+(k1-ka)ps.
As a stepping stone towards doing this, we will first prove ℓ≥ka. We use the following claim, which we will prove later:
ClaimWe have a decomposition
V=S1⊕…⊕Sℓ⊕T1⊕…⊕Tt
for some invariant subspaces S1,…,Sℓ,T1,…,Tt of V such that, after possible replacement of D (without changing ℓ),
Di=Si¯⊕Si′ 𝑎𝑛𝑑 Ej=Tj′,
where Si is a sum of indecomposable subspaces with distinct minimal polynomials for i=1,…,ℓ, and Tj is indecomposable for j=1,…,t.
Suppose by way of contradiction that ℓ<ka.
Certainly, then, either a=1 and ℓ<k1, or m>1, a=2 and ℓ<k2≤k1.
Hence, using the decomposition of V in the Claim, at most k1-1 indecomposables with minimal polynomial r1 appear in S1⊕…⊕Sℓ, and, when a=2, at most k2-1 indecomposables with minimal polynomial r2 also appear. But k1 and k2 copies of indecomposables with minimal polynomial r1 and r2, respectively, appear in the decomposition of V. Hence t≥a and, without loss of generality, T1 is indecomposable with minimal polynomial r1, and, in the case a=2, we may suppose T2 is indecomposable with minimal polynomial r2. Put
S={T1¯⊕T1′if a=1,T1⊕T2¯⊕(T1⊕T2)′if a=2,
where, in the second case, (T1⊕T2)′=T1′∩T2′=E1∩E2, which is indeed a complement for T1⊕T2. But
core(S)=E1, if a=1, and core(S)=E1∩E2, if a=2, so that the collection
𝒞′={𝒟∪{S}∪ℰ\{E1〈T〉}if a=1,𝒟∪{S}∪ℰ\{E1〈T〉,E2〈T〉}if a=2,
affords a faithful representation of G, but with degree less than the degree of the representation afforded by 𝒞, since
|G:S|=pq<aps={|G:E1〈T〉|if a=1,|G:E1〈T〉|+|G:E2〈T〉|if a=2.
This contradicts that 𝒞 is minimal. Hence ℓ≥ka.
There are at most ℓ occurrences of indecomposables with minimal polynomial r1 appearing in S1⊕…⊕Sℓ, so at least k1-ℓ such indecomposables must occur amongst T1,…,Tt, so that
t≥k1-ℓ.
Thus
ℓpq+tps=kapq+(ℓ-ka)pq+tps
≥kapq+(ℓ-ka)(a-1)ps+ps{0if a=1,k1-ℓif a=2,
=kapq+(k1-ka)ps
and (4.4) is proven. The statement of the theorem for s>1 is therefore captured succinctly by the formula
(4.5)μ(G)=kapq+(k1-ka)ps.
To complete the proof of the theorem, it therefore remains to verify the Claim. As a first step we prove
(4.6)V=T1⊕…⊕Tt⊕(E1∩…∩Et)
for some indecomposables Ti such that Ei=Ti′ for i=1,…,t. Note that we have V=E1⊕T1 for some indecomposable T1, so E1=T1′, which starts an induction.
Suppose, as inductive hypothesis, that for k≤t,
V=T1⊕…⊕Tk-1⊕(E1∩…∩Ek-1),
for some indecomposables T1,…,Tk-1 such that Ei=Ti′ for i=1,…,k-1. By the minimality of 𝒞, E1∩…∩Ek is a proper subspace of E1∩…∩Ek-1. Further,
E1∩…∩Ek-1E1∩…∩Ek≅(E1∩…∩Ek-1)+EkEk=VEk,
which is indecomposable, so we may choose an indecomposable Tk such that
E1∩…∩Ek-1=(E1∩…∩Ek)⊕Tk.
Certainly Tk is not a subspace of Ek (for otherwise E1∩…∩Ek∩Tk≠{0}), so it follows that V=Ek⊕Tk, so we may write Ek=Tk′. Then
V=(T1⊕…⊕Tk-1)⊕(E1∩…∩Ek-1)=T1⊕…⊕Tk⊕(E1∩…∩Ek),
which completes the inductive step and the proof of (4.6).
Note that if ℓ=0 (so that 𝒟=∅), then (4.6) proves the Claim (for then 𝒞=ℰ and E1∩…∩Et={0} so that V=T1⊕…⊕Tt).
We may suppose in what follows that ℓ>0.
Put E=E1∩…∩Et.
We next prove, by induction, that we can replace 𝒟 (if necessary) so that the following holds for k=0,…,ℓ:
(4.7)V=S1⊕…⊕Sk⊕T1⊕…⊕Tt⊕(S1′∩…∩Sk′∩E),
where Di=Si¯⊕Si′ and Si is a sum of indecomposables with distinct minimal polynomials, for i=1,…,k. This suffices to prove the Claim, because when k=ℓ we have
S1′∩…∩Sk′∩E=S1′∩…∩Sℓ′∩E=⋂𝒞={0}.
Note that (4.6) now becomes the initial case k=0 in a proof by induction of (4.7).
Suppose, as inductive hypothesis, that 0<k≤ℓ and we can replace 𝒟 (if necessary) so that
V=S1⊕…⊕Sk-1⊕T1⊕…⊕Tt⊕(S1′∩…∩Sk-1′∩E),
where Di=Si¯⊕Si′ and Si is a sum of indecomposables with distinct minimal polynomials for i=1…,k-1.
By the minimality of 𝒞,
core(D1∩…∩Dk-1∩E)≠core(D1∩…∩Dk∩E),
that is,
S1′∩…∩Sk-1′∩E≠S1′∩…∩Sk-1′∩E∩core(Dk).
But
S1′∩…∩Sk-1′∩ES1′∩…∩Sk-1′∩E∩core(Dk)≅(S1′∩…∩Sk-1′∩E)+core(Dk)core(Dk)
≤VcoreDk≅core(Dk)′,
which is a sum of indecomposables with distinct minimal polynomials. Hence
(S1′∩…∩Sk-1′∩E∩core(Dk))⊕Sk=S1′∩…∩Sk-1′∩E
for some invariant subspace Sk contained in E, which is a sum of indecomposables with distinct minimal polynomials. Choose any complement (S1′∩…∩Sk-1′∩E)′ and put
Sk′=(S1′∩…∩Sk-1′∩E∩core(Dk))⊕(S1′∩…∩Sk-1′∩E)′,
which is indeed a complement of Sk.
Put
Dk~=Sk¯⊕Sk′.
Observe that core(Dk~)=Sk′ and
S1′∩…∩Sk-1′∩E∩core(Dk~)
=S1′∩…∩Sk-1′∩E∩Sk′
=(S1′∩…∩Sk-1′∩E)∩[(S1′∩…∩Sk-1′∩E∩core(Dk))
⊕(S1′∩…∩Sk-1′∩E)′]
=S1′∩…∩Sk-1′∩E∩core(Dk),
so we may replace Dk by Dk~ in 𝒟 without disturbing faithfulness or the degree of the representation afforded by 𝒞. Renaming Dk~ by Dk, we get
V=S1⊕…⊕Sk-1⊕T1⊕…⊕Tt⊕(S1′∩…∩Sk-1′∩E)
=S1⊕…⊕Sk-1⊕T1⊕…⊕Tt
⊕((S1′∩…∩Sk-1′∩E∩core(Dk))⊕Sk)
=S1⊕…⊕Sk-1⊕T1⊕…⊕Tt⊕(Sk⊕(S1′∩…∩Sk-1′∩E∩Sk′))
=S1⊕…⊕Sk⊕T1⊕…⊕Tt⊕(S1′∩…∩Sk′∩E),
completing the inductive step, and (4.7) is proved. This completes the proof of the Claim and therefore also the proof of the theorem.
∎
Formula (4.5) captures the three alternatives in the previous theorem when s>1. However, by Remark 4.4 and Theorem 4.5, we have the following further simplification (eventually) if there turn out to be only finitely many generalised Mersenne primes:
Corollary 4.6With the hypotheses of Theorem 4.5, if s>1 and there are only finitely many generalised Mersenne primes, then there is an integer N such that for all q≥N, μ(V⋊T)=k1pq.
Example 4.7The smallest instance when q>ps-1, so that the third alternative of (4.1) is able to kick in, occurs when p=2 and q=3, so that s=2. Let T=[0111], so that φT=x2+x+1, and put G=𝔽22⋊T≅C22⋊C3≅Alt(4). As expected, (4.1) predicts correctly that μ(G)=ps=4.
Theorem 4.8Suppose that r1,…,rm are distinct irreducible polynomials over Fp of degree s≥2, where s is the order of p modulo q, such that
φT=(x-1)r1…rm 𝑎𝑛𝑑 χT=(x-1)kr1k1…rmkm.
We may suppose k1≥k2≥…≥km.
Then
(4.8)μ(V⋊T)={k1pqif k≤k1,q<ps-1,k1pq+(k-k1)pif k>k1,q<ps-1,k1ps+kpif m=1,q>ps-1,k2pq+(k1-k2)psif m>1,k≤k2,q>ps-1,k2pq+(k1-k2)ps+(k-k2)pif m>1,k>k2,q>ps-1.
Proof.
As before, let a be the smallest integer such that q<aps-1. By Lemma 4.3, a=1 or 2. We again put ka=0 when m=1 and a=2.
Put G=V⋊T=V〈T〉. We have a decomposition V=V~⊕Z, where
V~=⊕(i,j)∈IVij and Z=⊕α=1kZα,
where the Vij are indecomposable subspaces of V with minimal polynomials from amongst r1,…,rm, adopting the notation of the proof of the previous theorem, and the Zα are one-dimensional indecomposable subspaces of V on which the action of T is trivial (so Zα〈T〉≅Cp×Cq). By Theorem 4.5 and (4.5),
(4.9)μ(V~〈T〉)=kapq+ps(k1-ka).
Certainly, by (1.1), we have μ(G)≥μ(V~〈T〉). There are two cases.
Case 1: Suppose that ka≥k.
Let 𝒞 be the collection of subgroups described in the first part of the proof of Theorem 4.5 that affords a faithful representation of V~〈T〉 of degree μ(V~〈T〉), replacing V by V~ throughout. For α=1,…,k, put
Uα=Wα⊕Zα and Hα^=Uα¯⊕Wα′⊕⊕β≠αZβ,
where Uα¯ is a canonical codimension 1 subspace of Uα with trivial core (see Lemma 3.6), and here Wα′ denotes a complement of Wα in V~, so that
core(Hα^)=Wα′⊕⊕β≠αZβ.
Now put
𝒞^={H1^,…,Hk^,Hk+1⊕Z,…,Hka⊕Z}∪{Kj⊕Z∣(1,j)∈X},
where the notation Kj⊕Z represents the internal semidirect product resulting from joining Kj with Z (since the action of T on Z is trivial).
Then
core(⋂𝒞^)=core(⋂𝒞)⊕⋂α=1k⊕β≠αZβ={0},
so 𝒞^ affords a faithful representation of G. Its degree is the same as the degree of the representation of V~〈T〉 afforded by 𝒞, which is μ(V~〈T〉), so
μ(G)≤μ(V~〈T〉)≤μ(G),
whence we have equality. Formula (4.9) captures the first and fourth alternatives in
(4.8).
Case 2: Suppose that k>ka.
We make the same definitions as in the previous case, except that we put
𝒞^={H1^,…,Hka^}∪{Kj⊕Z∣(1,j)∈X}
∪{(V~⊕⊕β≠αZβ)〈T〉|α=ka+1,…,k}.
Again the representation of G afforded by 𝒞^ is faithful. Its degree is
kapq+(k-ka)p+ps(k1-ka),
which therefore serves as a lower bound for μ(G).
By Proposition 4.1, there exists a collection 𝒞=𝒟∪ℰ of subgroups affording a minimal representation of G such that
𝒟={D1,…,Dℓ} and ℰ={E1〈T〉,…,Et〈T〉},
where D1,…,Dk are codimension 1 subspaces of V and, after reordering (if necessary), E1,…,Et0 are complements of indecomposables with minimal polynomials from amongst r1,…,rm and Et0+1,…,Et are complements of one-dimensional indecomposables. As before, ℓ≥ka and, by the same reasoning as before, t0≥k1-ℓ
and t-t0≥k-ℓ.
By the definition of a, and since p≠q, we have (a-1)ps-1<q, so
pq≥(a-1)ps+p.
Hence
μ(G)=ℓpq+(t-t0)p+t0ps
=kapq+(ℓ-ka)pq+(t-t0)p+t0ps
≥kapq+(ℓ-ka)((a-1)ps+p)+(k-ℓ)p+ps{0if a=1,k1-ℓif a=2,
=kapq+(k-ka)p+ps(k1-ka),
whence we have
(4.10)μ(G)=kapq+(k-ka)p+ps(k1-ka).
Formula (4.10) captures the second, third and fifth alternatives in (4.8), and the proof is complete.
∎
Illustrations of formula (4.8) are implicit in applications in the next section.
5 Adding direct factors without increasing the degree
Results of the preceding section are applied now to investigate possible ways in which μ may fail to be additive with respect to taking direct products. The question of when additivity occurs is an important theme in the work of Johnson [11] and Wright [22]. The failure of additivity in general was demonstrated by a seminal example in [22] and explored further by Saunders [18, 17, 19]. In all their cases, nontrivial groups G and H are exhibited in which G does not decompose nontrivially as a direct product, H is a cyclic group of prime order and
(5.1)μ(G×H)=μ(G).
We reproduce these examples below as special cases of applications of the formulae in Theorems 4.5 and 4.8. By combining these formulae with Theorem 2.7, we finish by exhibiting examples of groups G that do not decompose nontrivially as direct products, but such that (5.1) holds for arbitrarily large direct products H of elementary abelian groups (with mixed primes).
Example 5.1Consider the groups
G1=𝔽52⋊T1,G2=𝔽53⋊T2,G3=𝔽54⋊T3,
where
T1=[0414],T2=[040140001],T3=[0400140000100001].
Then
|T1|=|T2|=|T3|=3,
φT1=χT1=x2+x+1,
φT2=χT2=φT3=(x-1)(x2+x+1),
χT3=(x-1)2(x2+x+1).
Then G1≅C52⋊C3 and μ(G1)=15, by the second alternative of (4.1). A minimal faithful representation is afforded by a canonical core-free subspace of 𝔽52 (see Lemma 3.6), yielding
G1≅〈a1,a2,b∣a15=a25=b3=1=[a1,a2],a1b=a2,a2b=a1-1a2-1〉
≅〈α1,α2,β〉,
where
α1=(12345)(678910)(1114121513),
α2=(12345)(697108)(1112131415),
β=(1116)(2127)(3138)(4149)(51510).
By the first alternative of (4.8), we have μ(G2)=15. A minimal faithful representation is afforded by a canonical core-free subspace of 𝔽53, yielding
G2≅G1×C5≅〈α1,α2,α3,β〉,
where α1, α2 and β are as above, and
α3=(12345)(678910)(1112131415).
In fact, G1 and G2 are isomorphic to subgroups of the transitive permutation group introduced at the end of Wright’s paper [22], which was the first published counterexample to additivity of μ with respect to direct product.
By contrast, now using the second alternative of (4.8), μ(G3)=15+5=20. A faithful intransitive representation of G3 is given by the previous canonical core-free subspace of 𝔽53, augmented in an obvious way in 𝔽54, and a subgroup of index 3, yielding
G3≅G2×C5≅G1×C52≅〈α1,α2,α3,α4,β〉,
where α1, α2, α3 and β are as above, but fixing five new letters, and
α4=(1617181920).
Observe that μ(C5)2=10, so that
(5.2)max{μ(G1),μ(C52)}=15<μ(G1×C52)=20<25=μ(G1)+μ(C52).
This answers affirmatively a question of Saunders [17], whether there exist groups K and L such that
(5.3)max{μ(K),μ(L)}<μ(K×L)<μ(K)+μ(L).
Note that if G and H are groups such that
μ(H)<μ(G) and μ(G×H)=μ(G)<μ(G)+μ(H)
(such as the example in [22]), then (5.3) holds easily by taking any group M of order coprime to |G×H|, putting K=G and L=M×H, and invoking Johnson’s result that μ is additive with respect to taking direct products of groups of coprime order. However, the solution (5.2) given here appears to be novel in that only two primes, namely 3 and 5, divide |K×L|, taking K=G1 and L=C52. This example clearly generalises, by (4.8), to an infinite class of examples, where (5.3) holds and only two distinct primes p and q divide |K×L|. Note that (5.3) fails, if K×L is a p-group, since μ is additive with respect to taking direct products of nilpotent groups by a theorem of Wright [22].
Example 5.2Let p and q be primes such that p has order s=q-1 modulo q, so that π=1+x+…+xq-1 is irreducible over 𝔽p. Suppose also (p,q)≠(2,3), as this guarantees that q<pq-2=ps-1, so that the second alternative of (4.1) will apply. (The case (p,q)=(2,3) is explored above in Example 4.7 when illustrating the third alternative of (4.1).) The smallest case satisfying our conditions is (p,q)=(2,5).
Consider the groups
H1=𝔽pq-1⋊T1≅Cpq-1⋊Cq and H2=𝔽pq⋊T2≅Cpq⋊Cq≅H1×Cp,
where T1 and T2 are matrices over 𝔽p in rational canonical form having characteristic polynomials π and (1+x)π respectively. Then
μ(H1)=pq=μ(H2)=μ(H1×Cp),
by the second alternative of (4.1) and the first alternative of (4.8).
Observe that H1 is a subgroup of the complex reflection group C(p,p,q), a member of the infinite class of counterexamples studied by Saunders in [18]. In the smallest case, when p=2 and q=5, the groups become
H1≅C24sdC5 and H2≅C25sdC5≅H1×C2,
and
μ(H1)=μ(H1×C2)=10<12=μ(H1)+μ(C2).
The group H1 and these properties appear for the first time in [17]. It is gratifying that the smallest example that comes from Saunders’ investigations, where he was motivated by questions about complex reflection groups, also coincides with the smallest example that arises as an application of Theorems 4.5 and 4.8. By results in [4], it is impossible to create a smaller example by any method, in the sense that G×H cannot embed in Sym(9) and have H nontrivial and μ(G)=μ(G×H).
We say that an integer m≥3 is Mersenne with respect to an integer n≥2 if m=1+n+…+nα for some integer α. Note that this implies
m=nα+1-1n-1<nα+1.
Lemma 5.3If m is Mersenne with respect to n, then k is not Mersenne with respect to n for m<k≤2m.
Proof.
If m and k are Mersenne with respect to n and m<k≤2m, then there exist α and β such that m=1+n+…+nα and k=m+nα+1+…+nα+β, whence nα+1≤nα+1+…+nα+β=k-m≤m<nα+1, which is impossible.
∎
The following corollary is of independent interest and probably well known.
Corollary 5.4Given a positive integer n, there exists infinitely many primes that are not Mersenne with respect to n.
Proof.
This follows quickly from Lemma 5.3 and Bertrand’s postulate.
∎
Lemma 5.5Let n≥2, k≥3 and N any positive integer. Then any strictly increasing sequence of k integers strictly between N and 2N contains a consecutive subsequence of ⌊k/2⌋ elements, none of which are Mersenne with respect to n.
Proof.
Let t1,…,tk be a strictly increasing sequence of integers strictly between N and 2N. If ti is not Mersenne with respect to n for all i, then we are done using the entire sequence. Suppose then that some element in the sequence is Mersenne with respect to n, and let tj be the least such element. Then, for all ℓ such that j<ℓ≤k, we have N<tj<tℓ<2N<2tj, so that tℓ is not Mersenne with respect to n, by Lemma 5.3. If j>⌊k/2⌋, then t1,…,t⌊k/2⌋ is a consecutive subsequence of ⌊k/2⌋ elements, none of which are Mersenne with respect to n, and we are done. Otherwise j≤⌊k/2⌋ and tj+1,…,tk is a consecutive subsequence with k-j≥k-⌊k/2⌋≥⌊k/2⌋ elements, none of which are Mersenne with respect to n, and again we are done.
∎
Proposition 5.6If p1,…,pk are prime numbers, then there exist infinitely many primes that
are not Mersenne with respect to pi for each i.
Proof.
Let p1,…,pk be primes and N any positive integer. By the Green–Tao theorem [7] there exists an arithmetic progression of primes
q-M,q-M+1,…,q0=q,q1,…,qM
for some M≥max{N,2k}. We may suppose the common difference is s so that q=q-M+Ms≥2ks and qi=q+is for each i=1,…,M. In particular,
(5.4)q<q1<…<qM<2q.
By Lemma 5.5, there exists a consecutive subsequence of q1,…,qM, starting at qi1 for some i1≥1, of length M1=⌊M/2⌋≥2k-1 consisting of elements none of which are Mersenne with respect to p1, which starts an induction. Suppose j≤k and, as inductive hypothesis, that we have a consecutive subsequence starting at qij-1 of length Mj-1≥2k-j+1 consisting of elements none of which are Mersenne with respect to p1,…,pj-1. By Lemma 5.5, this contains a consecutive subsequence starting at qij for some ij≥ij-1 of length Mj≥⌊Mj-1/2⌋≥2k-j consisting of elements none of which are Mersenne with respect to p1,…,pj, establishing the inductive step. The lemma now follows by induction by observing that Mk≥2k-k=1, so that we have found at least one prime qik≥N that is not Mersenne with respect to p1,…,pk.
∎
Remark 5.7Ramanujan [16] showed that π(n)-π(n/2) tends to infinity as n does, where π(n) denotes the number of primes less than or equal to n, generalising Bertrand’s postulate. This also guarantees the existence of an integer q and primes q1,…,qM such that (5.4) holds, and the proof of Proposition 5.6 proceeds as above, but avoiding use of the Green–Tao theorem.
In the following example, given an arbitrarily large direct product H of elementary abelian groups built from any collection of primes and positive integer exponents, we construct a group G such that μ(G×H)=μ(G), yet G does not decompose nontrivially as a direct product.
Example 5.8Let P={p1,…,pk} be a finite collection of distinct primes and N={n1,…,nk} a collection of positive integers. Choose a prime q≥5 that is not Mersenne with respect to each prime in P, and larger than all of the primes in P, the existence of which is guaranteed by Proposition 5.6.
Consider an integer i∈{1,…,k}. Let si be the multiplicative order of pi modulo q and put mi=sini. Then si>1 and we can find a monic irreducible polynomial πi∈𝔽pi of degree si such that its roots in an extension of 𝔽pi are primitive qth roots of 1.
We have q<pisi-1, by
Lemma 4.3, since q is not Mersenne with respect to pi.
Denote the companion matrix over a field 𝔽 of a monic polynomial π∈𝔽[x] by Mπ.
Define Ti to be the mi×mi matrix over 𝔽pi that is the matrix direct sum of ni copies of Mπi.
Now put
Ti^=Ti⊕Ini,
where Ini is an identity matrix (over 𝔽pi).
Then |Ti|=|Ti^|=q,
φTi=πi,χTi=πini,φTi^=(x-1)πi and χTi^=(x-1)niπini.
Now let Gi=Vi⋊Ti and Gi^=Vi^⋊Ti^, where
Vi=𝔽pimi and Vi^=𝔽pimi+ni.
Then
(5.5)μ(Gi)=μ(Gi^)=nipiqi,
by Theorems 4.5 and 4.8.
Observe that, because Ini acts trivially on 𝔽pini,
(5.6)Gi^≅Gi×Cpini.
Now put
T=⊕i=1kTi,T^=⊕i=1kTi^,V=⊕i=1kVi,V^=⊕i=1kVi^,
where the zeros outside the matrix blocks down the diagonals act as formal zeros (not in any particular field) for the purpose of matrix multiplication, and the elements of V and V^ may be regarded as column vectors over 𝔽p1∪…∪𝔽pk. Thus, because the construction respects direct sum decompositions, T and T^ may be regarded as acting on V and V^ (on the left) by usual matrix multiplication. Hence, as in (3.1) and (3.2), we may define
G=V⋊T and G^=V^⋊T^.
The actions of T and T^ on the respective ith direct summands is nontrivial, for each i, and the orders of these direct summands are pairwise coprime and also coprime to q, so, by repeated application of the last alternative in the formula given in Theorem 2.7 and by (5.5), we have
μ(G)=∑i=1kμ(Gi)=∑i=1knipiqi=∑i=1kμ(Gi^)=μ(G^).
Also, by (5.6),
G^≅G×Cp1n1×…×Cpknk.
Finally, put H=Cp1n1×…×Cpknk, which is our arbitrarily large direct product of elementary abelian groups, using all of the primes p1,…,pk. Then μ(G×H) equals μ(G). By construction, the irreducible action on each direct summand guarantees that G does not decompose nontrivially as a direct product.
Note that when H=Cp1×…×Cpk, the action of G on each Sylow pi-subgroup is irreducible. The authors are not aware of any simpler method for achieving this last property, which appears to be inextricably linked to number-theoretic properties of the particular primes involved.
Remark 5.9It is an open problem whether there are finitely many generalised Mersenne primes. If there are only finitely many, then we can avoid the use of Proposition 5.6 in the previous example, simply by choosing q to be larger also than the largest generalised Mersenne prime (for that would guarantee q<pisi-1 for each i, by Remark 4.4).
and
Michael Hendriksen
