Finite nearfields classified, again

Theorem 1.1.

Let G be a finite subgroup of GL⁢(n,C) and let R be a Sylow r-subgroup of G for some prime r.

1. If R contains an element of order at least rn, then G has a nontrivial normal r-subgroup.

2. If |R|=r≥4⁢n-2, then R is normal in G.

Proof.

Up to conjugation G is a subgroup of GL⁢(n,A), where A is the ring of all algebraic integers in a finite extension field L of ℚ; see [9, V.11.7 (a) and V.12.5 (b)], [4, Corollary 3.8 and Corollary 3.4A] or [17, 15.11]. We choose a maximal ideal I of A with r∈I. Then the field A/I has characteristic r.

(i) Reduction modulo I yields a homomorphism of G into GL⁢(n,A/I). We claim that the kernel K of this homomorphism is an r-group. Otherwise K contains an element g of prime order s≠r. The ring A is noetherian, hence we have ⋂m≥1Im={0} by Krull’s intersection theorem. For some integer m≥1 the ideal Im contains all entries of the matrix g-1, but Im+1 does not. Then

Thus Im+1 contains all entries of s⁢(g-1), and hence all entries of g-1 by Bezout’s lemma. This is a contradiction. (See also [4, Theorem 3.6B].)

If K were trivial, then GL⁢(n,A/I) would contain an element g of order rt≥r⁢n. The minimal polynomial of g divides xrt-1=(x-1)rt and has degree at most n≤rt-1. Thus (g-1)n=0 and grt-1-1=(g-1)rt-1=0, hence g has order at most rt-1; this is a contradiction. Therefore K is a nontrivial normal r-subgroup of G. (We have solved Exercise 2 (a) in [4, p. 94].)

(ii) The following arguments evolved from the proof of Robinson [20, Theorem A]; his proof uses character theory and tensor products like V⊗V∗ and V⊗V∗⊗V⊗V∗, which are naturally isomorphic to the endomorphism rings End⁡V and End⁡End⁡V, respectively.

Let V=Ln, considered as a vector space over L (or let V=ℂn). We write the action of G on End⁡V by conjugation as g2⁢(f):=g⁢f⁢g-1 for g∈G, f∈End⁡V, and the action on End⁡End⁡V as g4⁢(ϵ):=g2⁢ϵ⁢g2-1 for ϵ∈End⁡End⁡V. Then G2:={g2∣g∈G}≤GL⁢(End⁡V)⊆End⁡End⁡V. Let H:=NG⁢R be the normalizer of R in G.

We claim:

1. The centralizers of G2 and H2 in End⁡End⁡V coincide.

This is the crucial step, which uses the assumptions on r. We have H2≤G2, hence CEnd⁡End⁡V⁢G2⊆CEnd⁡End⁡V⁢H2. We assume that this inclusion is strict and aim for a contradiction. Define ψ:CEnd⁡End⁡V⁢H2→CEnd⁡End⁡V⁢G2 by

where T is any left transversal of H in G, i.e. G is the disjoint union ⋃t∈Tt⁢H; note that (t⁢h)4⁢(ϵ)=t4⁢(h4⁢(ϵ))=t4⁢(ϵ) for h∈H and

The subring B:={x⁢y-1∣x∈A,y∈A∖I} of L is the localization of A at I; thus B is a discrete valuation ring, and its unique maximal ideal is of the form bB with b∈B. The ring M:=EndB⁡(EndB⁡Bn) is a B-module, and G2 is a subset of M as G≤GL⁢(n,A)≤GL⁢(n,B). The nontrivial kernel of the L-linear map ψ is described by a system of homogeneous linear equations with coefficients from A⊆B. Thus we find a nonzero element m∈CM⁢H2 with ψ⁢(m)=0, and we may assume that m∉b⁢M by dividing m by a suitable power of b.

Let R=〈z〉 and let G=H∪⋃d∈DR⁢d⁢H be the double coset decomposition of G with respect to (R,H), with a suitable subset D⊆G∖H. Then the set T:={1}∪{zi⁢d∣0≤i≤r-1,d∈D} is a left transversal of H in G, because d-1⁢zi-j⁢d∈H=NG⁢R and zi-j≠1 would imply d∈H, as |R|=r and R=〈zi-j〉. We obtain

In the next paragraph we show that ∑i=0r-1z4i maps M into bM; this gives the contradiction m∈b⁢M.

We reduce modulo b. The quotient Bn¯:=Bn/b⁢Bn is a vector space of dimension n over the field B/b⁢B, which has characteristic r. Let z¯∈End⁡(Bn¯) be the linear map induced by z. As zr=1, we have (1-z¯)r=1-z¯r=0, which implies (1-z¯)n=0 (since the minimal polynomial of 1-z¯ has degree at most n). Now z¯2=λ⁢ρ-1 where λ,ρ∈End⁡End⁡(Bn¯) denote the left and right multiplications with z¯, respectively. Since λ and ρ commute, we obtain

as every summand in the binomial expansion of ((1-λ)+(ρ-1))2⁢n-1 involves a factor equal to 0. We repeat this argument:

where now λ,ρ∈End⁡End⁡End⁡(Bn¯) denote the left and right multiplications with z¯2, respectively. Then

In view of r≥4⁢n-2 we obtain in End⁡End⁡End⁡(Bn¯) the equation

as (r-1i)≡(-1)imodr. Hence ∑i=0r-1z4i∈EndB⁡M maps M into bM. This completes the proof of assertion (∗).

We recall the double centralizer theorem: if E is a finite-dimensional vector space over a field and X is a completely reducible subalgebra of End⁡E, then X=CEnd⁡E⁢(CEnd⁡E⁢X); see e.g. [11, Theorem 5.1, p. 258, or Theorem 4.10, p. 222]. By Maschke’s theorem, the finite groups G2 and H2 generate completely reducible L-subalgebras of End⁡End⁡V. Assertion (∗) and the double centralizer theorem imply that G2 and H2 generate in fact the same L-subalgebra of End⁡End⁡V. As R is normal in H, the centralizer CEnd⁡V⁢R is invariant under H2, hence also under G2. Thus G acts by conjugation on CEnd⁡V⁢R.

Let K be the kernel of this action of G on CEnd⁡V⁢R. Then CEnd⁡V⁢R⊆CEnd⁡V⁢K. Thus we have K⊆CEnd⁡V⁢(CEnd⁡V⁢K)⊆CEnd⁡V⁢(CEnd⁡V⁢R), which coincides with the L-algebra generated by R by the double centralizer theorem. Since R is commutative, we infer that K is commutative. Hence the Sylow group R is characteristic in K and therefore normal in G. ∎

Theorem 1.2.

Let n≥3 and let p be a prime with p≥5. If G≤GL⁢(n,p) has order |G|=pn-1, then G is a subgroup of Γ⁢L⁢(1,pn), embedded naturally into GL⁢(n,p).

Proof.

By a result of Bang (1886) and Zsigmondy (1892) there exists a Zsigmondy prime r for pn-1, i.e. a prime r dividing pn-1 but not pk-1 for 1≤k<n; see e.g. [7, 3.9], [18, Corollary 1.2] or [14, II.6.2] and note that these primes r are often called p-primitive prime divisors of pn-1. Then n is the multiplicative order of p modulo r; in particular, n divides r-1, hence r≡1modn and r≥n+1.

Every subgroup Cr of order r in GL⁢(n,p) is irreducible, and the subring of the matrix ring 𝔽pn×n generated by Cr is isomorphic to the field 𝔽pn. The normalizer of Cr in GL⁢(n,p) is the group Γ⁢L⁢(1,pn), embedded naturally into GL⁢(n,p). Thus it suffices to show that G contains a normal subgroup of order r.

The Sylow r-subgroups of G are cyclic, since GL⁢(n,p) contains cyclic (Singer) subgroups of order pn-1. Moreover, G is isomorphic to a subgroup of GL⁢(n,ℂ) as p does not divide |G|: in fact, Serre [21, Theorem 4.25] obtains an inclusion G≤GL⁢(n,ℤp) from (a special case of) the Schur–Zassenhaus theorem, and the ring ℤp of p-adic integers is a subring of ℂ; see also [4, Corollary 3.8], [17, 15.11], [22, Satz 206] or [9, V.11.7a)]. Hence G contains a normal subgroup of order r by Theorem 1.1 if r2∣pn-1 or r≥4⁢n-2.

It remains to consider the situation where all Zsigmondy primes r for pn-1 satisfy r2∤pn-1 and r∈{n+1,2⁢n+1,3⁢n+1}. Then Φn⁢(p) divides the product n′⁢(n+1)⁢(2⁢n+1)⁢(3⁢n+1), where Φn is the nth cyclotomic polynomial and n′ is the largest prime divisor of n, see [7, 3.5]. The prime powers pn occurring here can be determined by elementary number theory, using the methods from [7, Section 3] or [18, Section 1]. In fact, this problem has been solved by Glasby, Lübeck, Niemeyer and Praeger [6], and we infer from [6, Theorem 3 and Table 5] that pn is one of the prime powers 54,56,176. (So far, we have used only that |G| is divisible by Φn⁢(p) and not divisible by p, but not yet the equation |G|=pn-1.)

For pn=54 we have |G|=24⋅3⋅13 and r=13. By Sylow’s theorem G has a normal subgroup of order 13.

For pn∈{56,176} the prime r=7 is a Zsigmondy prime for |G|, hence G is irreducible. Let s be the largest prime divisor of |G|=p6-1; thus s∈{31,307} and s is a Zsigmondy prime for p3-1. Since s2∤|G|, there exists a normal subgroup S of order s in G by Theorem 1.1 (ii). The subring of 𝔽p6×6 generated by S is normalized by the irreducible group G and therefore isomorphic to 𝔽p3. Thus G≤Γ⁢L⁢(2,p3), the centralizer CG⁢S=G∩GL⁢(2,p3) is normal in G with index at most 3, and |CG⁢S| is divisible by 7, but not by 72. As above, CG⁢S is isomorphic to a subgroup of GL⁢(2,ℂ). Since 7≥4⋅2-2, we infer from Theorem 1.1 (ii) that CG⁢S has a characteristic subgroup of order 7 (this follows also from a result of Blichfeldt, see [4, Theorem 5.5], as well as from Dickson’s list of all subgroups of PSL⁢(2,p3), see [13, XI.2.3], [9, II.8.24] or [23, 3.6.17]). This subgroup of order 7 is normal in G. ∎

Lemma 2.1.

A finite nearfield N is a Dickson nearfield if its multiplicative group N× has a normal 2-complement.

Proof.

Let C be a normal 2-complement in N×. All Sylow subgroups of C are cyclic, see [17, 18.1] or [19, 10.5.5]. The structure of such groups C is well known, see [19, 10.1.10] or [9, IV.2.11]; in particular, the commutator group C′ and C/C′ are cyclic, hence N× is supersolvable. Let A be maximal among the abelian normal subgroups of N×. Then A coincides with its centralizer CN×⁢(A); otherwise the supersolvable group CN×⁢(A)/A is not trivial, hence it has a nontrivial cyclic normal subgroup, which gives a contradiction to the choice of A.

We identify A with {x↦a⋅x∣a∈A}≤Aut⁡(N,+) and consider a minimal A-invariant subgroup U≠{0} of (N,+); then U is A-irreducible. A acts regularly on U∖{0}, hence |A|≤|U|-1 and the restriction A→A|U to U is injective. Moreover, A|U is contained in the multiplicative group of a (skew) field by Schur’s lemma. Thus A≅A|U is cyclic and therefore

We assume that U≠N and derive a contradiction. We have U≠b⋅U for some b∈N×. Since b⋅U is A-invariant, U∩b⋅U={0} and U⊕b⋅U is a subgroup of (N,+). Thus

which is a contradiction to |A|≤|U|-1.

Thus U=N, which means that A acts irreducibly on (N,+). Hence the subring F of End⁡(N,+) generated by A is a field by Schur’s lemma. Moreover, dimF⁡N=1 and F is normalized by G:={x↦c⋅x∣c∈N×}, hence we have G≤Γ⁢L⁢(1,F) and N is a Dickson nearfield. ∎

Corollary 2.2.

Every finite nearfield N with |N|≡0mod2 or |N|≢1mod3 is a Dickson nearfield.

Proof.

If |N| is even, then N× is a normal 2-complement in N× and Lemma 2.1 applies.

If |N|≢1mod3, then N× contains no element of order 3. Let S be any 2-subgroup of N×. By [17, 18.1] or [19, 10.5.5], S is cyclic or a generalized quaternion group. Hence either Aut⁡S is a 2-group, or S is the quaternion group of order 8 and Aut⁡S≅S4 has order 24; see [17, 9.9, 9.10]. Thus S is centralized by all elements of odd order in its normalizer in N×. By a result of Frobenius, see [19, 10.3.2] or [17, 13.3 or 13.4], N× has a normal 2-complement and Lemma 2.1 applies again. ∎

Lemma 2.3.

Let H be a group containing only one element c of order 2.

1. If H/〈c〉≅A4, then H≅SL⁢(2,3).

2. If H/〈c〉≅S4, then H≅2-⁢S4.

3. If H/〈c〉≅A5, then H≅SL⁢(2,5).

Proof.

The first two assertions are proved in [10, XII.8.5]; we consider the third assertion. Since SL⁢(2,5) satisfies the assumptions, it suffices to prove that the isomorphism type of H is uniquely determined.

The group A5 has the presentation 〈x,y∣x2=y3=(x⁢y)5=1〉, hence H contains elements a and b with a2,b3,(a⁢b)5∈〈c〉 and H=〈a,b,c〉. We infer that a2=c, and we may assume that b3=1=(a⁢b)5, multiplying b or a by c if necessary. The group H~ with the presentation

admits an epimorphism H~→H mapping x,y,z to a,b,c in this order. The quotient H~/〈z〉 has the presentation 〈x,y∣x2=y3=(x⁢y)5=1〉≅A5. Hence |H~|=2⋅|A5|=|H| and H is isomorphic to H~. (Similar arguments work for the groups A4 and S4 with the presentations 〈x,y∣x2=y3=(x⁢y)3=1〉 and 〈w,x,y∣w2=x2=y2=(w⁢x)3=(x⁢y)3=(w⁢x)2=1〉.) ∎

Theorem 2.4 (Zassenhaus).

Every finite nearfield N is either a Dickson nearfield, or |N|=p2 and one of the following holds:

1. p=5 and N×≅SL⁢(2,3),

2. p=7 and N×≅2-⁢S4,

3. p=11 and N×≅SL⁢(2,3)×C5,

4. p=11 and N×≅SL⁢(2,5),

5. p=23 and N×≅2-⁢S4×C11,

6. p=29 and N×≅SL⁢(2,5)×C7,

7. p=59 and N×≅SL⁢(2,5)×C29.

In each of these seven cases, there exists precisely one isomorphism type for N.

Proof.

The group G:={x↦a⋅x∣a∈N×}≤Aut⁡(N,+) is isomorphic to N× and sharply transitive on N∖{0}. It follows that (N,+) is an elementary abelian p-group for some prime p, hence we have (N,+)=(𝔽pn,+) for some n∈ℕ and G≤Aut⁡(N,+)=GL⁢(n,p) has order |G|=pn-1. For p≠2 every involution in GL⁢(n,p) is diagonalizable, hence -id is the only involution in G.

If p∈{2,3}, then N is a Dickson nearfield by Corollary 2.2, hence we may assume that p≥5. If n=1, then N≅𝔽p since G=GL⁢(1,p)≅𝔽p×. If n≥3, then N is a Dickson nearfield by Theorem 1.2.

It remains to consider the case n=2. Let C:=G∩𝔽p×⁢id. If G¯:=G/C≤PGL⁢(2,p) has a cyclic subgroup of index 1 or 2, then G≅N× has an abelian subgroup of index 1 or 2; thus N× has a normal 2-complement, and N is a Dickson nearfield by Lemma 2.1. In the remaining cases G¯ is neither cyclic nor a dihedral group. We observe that PGL⁢(2,p)≤PSL⁢(2,p2), because every element of 𝔽p is a square in 𝔽p2. Since |G| is coprime to p, we infer that G¯ is isomorphic to A4, S4 or A5 from Dickson’s enumeration of all subgroups of PSL⁢(2,p2); see [13, XI.2.3], [9, II.8.24] or [23, 3.6.17]. We have 〈-id〉≤C≤𝔽p×⁢id, hence p+1⁢∣|G¯|∣⁢12⁢(p2-1), which leads to the following cases:

1. G¯≅A4 and p∈{5,11},

2. G¯≅S4 and p∈{7,23},

3. G¯≅A5 and p∈{11,19,29,59}.

In each case we have |C|=(p2-1)/|G¯|≡2(mod4), hence C=〈-id〉×Ct with a subgroup Ct≤𝔽p×⁢id of odd order t. If p=19, then we have t=3 and A5≅G¯=G¯≅′G′/G′∩C, hence G′≤SL⁢(2,19) contains an element g of order 3, with eigenvalues λ,λ-1∈𝔽19; thus Ct=〈λ⁢id〉 and g⁢λ⁢id∈G∖{id} has the eigenvalue 1, which is absurd. This eliminates the possibility p=19.

Let H:=G∩SL⁢(2,p). We claim that G/C=H⁢C/C. Both A4 and A5 have no proper normal subgroups with index dividing p-1=|GL(2,p):SL(2,p)|, hence G/C=H⁢C/C if G¯≅A4,A5. For G¯≅S4 we use the fact that -id∈C is the only involution in G. This implies that every involution in G¯ is of the form gC with g∈G and g2=-id, in view of g⁢C=gt⁢C. As g∉C, the polynomial x2+1 is the minimal polynomial and also the characteristic polynomial of g, hence det⁡g=1. Since S4 is generated by its involutions, this shows that G/C=H⁢C/C in all cases. We have H∩C=〈-id〉 and therefore

By Lemma 2.3 we have one of cases (i)–(vii).

In each case there exists at most one isomorphism type for N, since PGL⁢(2,p) contains only one conjugacy class of subgroups isomorphic to G¯. This is proved in [13, XI Appendix, Theorem 2, p. 202] and [2, 8.8, 8.14].

In fact, the group PSL⁢(2,p) contains a subgroup isomorphic to G¯ in each case; see [21, Section 10.2], [9, II.8.13, II.8.18], [13, XI Appendix, Theorem 1, p. 201] or [23, 3.6.26]. The preimage H≤SL⁢(2,p) of such a subgroup is fixed-point-free on 𝔽p2∖{0} since |H| is coprime to p. Hence H×Ct is fixed-point-free on 𝔽p2∖{0} for every subgroup Ct≤𝔽p×⁢id of order t coprime to |H|. For t=12⁢(p2-1)/|G¯| the group H×Ct is sharply transitive on 𝔽p2∖{0}. This implies the existence of N in each case. ∎

Corollary 3.1.

If G is a finite sharply two-transitive permutation group of degree q, then q is a prime power and up to permutation isomorphism one of the following holds:

1. G=AGL⁢(1,N)≤A⁢Γ⁢L⁢(1,q) for a finite Dickson nearfield N, and G is abelian-by-metacyclic.

2. q∈{52,72,112,232,292,592} and G=AGL⁢(1,N) for one of the seven nearfields N in Theorem 2.4.

Remark.

In [10, XII.9.5 Remark] the group GL⁢(2,3) appearing in cases b) and d) should be replaced by 𝔊48≅2-⁢S4 as defined in [10, XII.8.4], since GL⁢(2,3) contains several involutions.