Elsevier

Journal of Algebra

Volume 586, 15 November 2021, Pages 643-717
Journal of Algebra

Regular orbits of quasisimple linear groups II

https://doi.org/10.1016/j.jalgebra.2021.07.005Get rights and content

Abstract

Let V be a finite-dimensional vector space over a finite field, and suppose GΓL(V) is a group with a unique subnormal quasisimple subgroup E(G) that is absolutely irreducible on V. A base for G is a set of vectors BV with pointwise stabiliser GB=1. If G has a base of size 1, we say that it has a regular orbit on V. In this paper we investigate the minimal base size of groups G with E(G)/Z(E(G))PSLn(q) in defining characteristic, with an aim of classifying those with a regular orbit on V.

Introduction

A permutation group GSym(Ω) is said to have a regular orbit on Ω if there exists ωΩ with trivial stabiliser in G. The study of regular orbits arises in a number of contexts, particularly where Ω is a vector space V, and GGL(V). For instance, if V is a finite-dimensional vector space over a finite field, GGL(V), and all orbits of G on V{0} are regular, then the affine group GV is a Frobenius group with Frobenius complement G, and such G were classified by Zassenhaus [41]. Regular orbits also underpin parts of the proof of the k(GV)-conjecture [14], asserting that the number of conjugacy classes k(GV) of GV, with |G| coprime to |V|, is at most |V|. One of the major cases was where G is almost quasisimple and acts irreducibly on V. In this case, the existence of a regular orbit of G on V was sufficient to prove the k(GV)-conjecture. A classification of pairs (G,V) where G has a regular orbit on V and (|G|,|V|)=1 was completed by Köhler and Pahlings [26], building on work of Goodwin [15], [16] and Liebeck [30].

A subset BV is a base for G if its pointwise stabiliser in G is trivial. The minimal size of a base for G is called the base size and we denote it by b(G). For example, if G has a regular orbit on V, then b(G)=1. Each element gG is characterised by its action on a base, so b(G)log|G|/log|V|. In recent years, there has been a number of advancements towards classifying finite primitive groups H with small bases. A primitive group with b(H)=1 is cyclic, so the smallest case of interest is where b(H)=2. There have been a number of contributions to a classification of such groups, including a partial classification for diagonal type groups [9], a complete classification for primitive actions of Symm and Altm [5], [22] and sporadic groups [7] and also substantial progress for almost simple classical groups [6].

A finite group G is said to be quasisimple if it is perfect and G/Z(G) is a non-abelian simple group. We further define G to be almost quasisimple if G has a unique quasisimple subnormal subgroup, which forms the layer E(G) of G, and the quotient G/F(G) of G by its Fitting subgroup F(G) is almost simple. The Fitting subgroup of an almost quasisimple group GΓL(V) is generated by the group of scalars F0(G)Z(GL(V)) contained in G, as well as the set of field automorphisms in G that commute with E(G). A primitive affine group H=GV has base size 2 if and only if GGL(V) has a regular orbit in its irreducible action on V. In classifying G with a regular orbit, one would naturally use Aschbacher's theorem [1] to determine the possibilities for irreducible subgroups of GL(V). In this paper, we investigate the case where G is a member of the C9 class of Aschbacher's theorem. Slightly more broadly than this, we consider GΓL(V) almost quasisimple such that E(G) acts absolutely irreducibly on V.

The pairs (G,V) where G has a regular orbit on V have been classified for E(G)/Z(E(G)) a sporadic or alternating group [10], [11], and for G of Lie type with (|G|,|V|)=1 in the aforementioned proof of the k(GV)-conjecture. The authors of [29] showed that b(G)6 for GGL(V) with E(G) quasisimple, as long as V is not the natural module for E(G). Moreover, Guralnick and Lawther [18] classified (G,V) where G is a simple algebraic group over an algebraically closed field of characteristic p>0 that has a regular orbit on the irreducible G-module V. They also determine the generic stabilisers in each case. Their methods, which provide the foundation for the techniques in this paper, rely heavily on detailed analyses of highest weight representations of these simple algebraic groups.

This paper is the second in a series of three, which analyse base sizes of pairs (G,V), where GΓL(V) is almost quasisimple and E(G) is a group of Lie type that acts absolutely irreducibly on V, with (|G|,|V|)>1. The first paper [28] dealt with the cross-characteristic representations of G, while the present paper considers the case where E(G)/Z(E(G))PSLn(q) and V is an absolutely irreducible module for E(G) in defining characteristic. The final paper in the series will consider the remaining almost quasisimple groups GΓL(V) of Lie type in defining characteristic.

We say that two representations ρ1,ρ2 of E(G) (and their corresponding modules) are quasiequivalent if there exists gAut(E(G)) such that ρ1 is equivalent to gρ2. For an almost quasisimple group of Lie type G=G(q), and a natural number i, we define ϵi=1 if i|q, and ϵi=0 otherwise.

The main theorem of this work is as follows.

Theorem 1.1

Let V=Vd(q0) be a d-dimensional vector space over Fq0 with q0=pf, and let GΓL(V) be an almost quasisimple group such that E(G)/Z(E(G))PSLn(q) with n2 and q=pe. Suppose that the restriction of V to E(G) is an absolutely irreducible module V(λ) of highest weight λ. Set k=f/e. Either G has a regular orbit on V, or up to quasiequivalence, λ and n appear in Table 1.1 if k=1 and Table 1.2 otherwise.

There are several corollaries that we can deduce from Theorem 1.1.

Corollary 1.2

Let G, V be as in Theorem 1.1. If n7 and logq|V|>n2+n, then G has a regular orbit on V.

Note that the lower bound of n2+n in Corollary 1.2 is sharp only if G (as in Theorem 1.1) has no regular orbit on V=V(2λ1) when k=2. The authors of [29] show that if G as in Theorem 1.1 is contained in GL(V), then b(G)6, unless V is the natural module for E(G). The next corollary gives an improvement to this result.

Corollary 1.3

Let G, V be as in Theorem 1.1. Then either V is the natural module for G, or b(G)5.

We compute b(PSL4(2).2)=5 on V(λ2) over F2 using GAP [13], so the upper bound in Corollary 1.3 is sharp.

We now give some remarks on the content of Table 1.1, Table 1.2.

Remark 1.4

  • (i)

    The notation λi in Table 1.1, Table 1.2 denotes the ith fundamental dominant weight of the root system associated with G, labelled in the usual way. Moreover, the coefficients pa have 1ae1.

  • (ii)

    The rows in Table 1.1, Table 1.2 for λ=λ1 are asterisked because if G/F0(G) contains no field automorphisms, then n/kb(G)n/k+1, and b(G)=n/k+1 only if either i=(k,n)>1 and G contains scalars in Fqi×Fq×, or G contains field automorphisms.

  • (iii)

    Notice that if G has no regular orbit on V, then the possibilities for the parameter k are very restricted. Indeed, by Table 1.1, Table 1.2, excluding the natural module for E(G), the parameter k is either an integer k[1,4], or can be written as k=1/c, with c[2,4].

  • (iv)

    The entry for λ=3λ1 and n=2 is asterisked because, by Proposition 5.15, G has a regular orbit on V if G is quasisimple. Otherwise, b(G)2.

  • (v)

    The entries for λ=λ3 with n=6 in Table 1.1, Table 1.2 are asterisked because by Proposition 5.14, Proposition 8.13, 2/kb(G)(3+δ)/k, where δ is 1 if G contains a graph automorphism and zero otherwise.

  • (vi)

    The row for λ=λ1+λl in Table 1.2 is asterisked because by Proposition 8.4, there is a regular orbit of G on V if G does not contain graph automorphisms, and otherwise b(G)2.

  • (vii)

    If λ=(pa+1)λ1 or λ1+paλn1, then by Proposition 6.7, there is no regular orbit of G on V unless (a,e)=1, (n,p)=(2,2) or (2,3) and G=SL2(q)/K, where K is the kernel of the action of SL2(q).

  • (viii)

    The entries in the lower section of Table 1.1 are asterisked because if G is quasisimple, then G has a regular orbit on V. Otherwise, we determine that b(G)2.

  • (ix)

    Each row in Table 1.2 with λ=(i=0m1pie/m)λ1 for some m{2,3,4} describes an SLn(q)-module over a subfield Fq1/mFq. In each of these cases, k=1/m. The construction of such modules is described in Section 7.

  • (x)

    The papers [10], [11], [15], [16], [26] mentioned in the introduction each assume that V (as in Theorem 1.1) is a faithful d0-dimensional Fr0G-module, with r0 prime such that E(G) acts irreducibly, but not necessarily absolutely irreducibly, on V. We can reconcile this with our study of absolutely irreducible modules for E(G) over finite fields of arbitrary prime power order, following [15, §3]. Define k=EndFr0G(V), K=EndFr0E(G)(V), t=|K:k| and d=dimK(V). Then E(G)GLd(K) is absolutely irreducible, and GGLd(K)ϕΓLd(K), where ϕ is a field automorphism of order t.

If q=q0, that is, k=1, and G is quasisimple, we can say precisely when G has a regular orbit on V.

Corollary 1.5

Let G, V be as in Theorem 1.1, and further suppose G is quasisimple. If q=q0, then either G has a regular orbit on V, or b(G)>1 and V appears in Table 1.1.

Our main approach for proving Theorem 1.1 relies on the simple observation that if G has no regular orbit on the absolutely irreducible E(G)-module V, then every vector vV is fixed by a prime order element in G. For each V, we then set out to give a proof by showing that |V| exceeds the sum of the sizes of the fixed point spaces CV(x) of prime order elements xG1 (this is formalised in Proposition 3.1). This technique requires a method of determining reasonably precise upper bounds for the dimensions of fixed point spaces of elements of G.

To compute these upper bounds, we adopt the techniques pioneered by Guralnick and Lawther in [18], and also based on the work of Kenneally [24]. Their methods rely heavily on weight theory for representations of simple algebraic groups. They obtain upper bounds on eigenspace dimensions by defining a set of equivalence relations on the weights of the module V. These equivalence relations are derived from subsystems Ψ of the root system Φ associated to the ambient simple algebraic group G corresponding to E(G). Larger subsystems Ψ generally give tighter upper bounds, but apply to fewer conjugacy classes due to our underlying assumptions. The technique is described in more detail in Section 2.

The techniques implemented by Guralnick and Lawther provide a starting point for the proof of Theorem 1.1. However, our application to finite groups presents additional challenges. We are also required to sum over conjugacy classes of our group, and this often means that more delicate upper bounds on dimensions of fixed point spaces are required. In some cases, this method does not work at all, and different considerations are needed.

Remark 1.6

There are striking examples where Guralnick and Lawther [18] show that G, a simple algebraic group of type Al, has no regular orbit on an irreducible FqG-module V=V(λ), but according to Theorem 1.1, the corresponding finite group G(q) has a regular orbit on V realised over Fq, where q=pe. For example, when l=1, V=V((pa+1)λ1) and (a,e)=1, then [18, Proposition 3.1.8] asserts that there is no regular orbit under the action of SL2(Fq), but we prove in Proposition 6.7 that there is a regular orbit on V under the action of SL2(q) when p=2 or 3.

The rest of this work is set out as follows. In Section 2, we present some preliminary results, which will provide the machinery for the bulk of the proofs of the auxiliary results which together will prove Theorem 1.1. We also include some explanation on the techniques of the proofs and a guide on how information is presented in tables preceding each calculation. The proof of Theorem 1.1 is split across four sections. In the notation of Theorem 1.1, the modules V=V(λ) where k=1 are dealt with in Section 5 for λ p-restricted and Section 6 otherwise. Section 7 deals with absolutely irreducible modules with k<1 that are not realised over a proper subfield of Fqk. Finally, Section 8 completes the proof of Theorem 1.1 by considering field extensions of the modules discussed in Sections 5, 6 and 7.

Section snippets

Background

Let G be a simple algebraic group over Fp, p prime. Let TG be a fixed maximal torus of G, and Φ be the root system of G with respect to T, with base Δ={α1,,αl} of simple roots, and corresponding fundamental dominant weights λ1,,λl. We define a partial ordering on weights λ,μ by saying μλ if and only if λμ is a non-negative linear combination of simple roots. By a theorem of Chevalley, the irreducible modules of G in defining characteristic p are characterised by their unique highest

Techniques

The following proposition gives the main results used to prove Theorem 1.1. For an almost quasisimple group G, let Gs denote the set of elements of G of projective prime order s, and Gs denote the set of elements of G with projective prime order coprime to s.

Proposition 3.1

[28, Proposition 3.1]

Let GΓL(V) be an almost quasisimple group, acting irreducibly on the d-dimensional module V=Vd(r) over Fr. Set H=G/F(G) and let P be a set of conjugacy class representatives of elements of projective prime order in G. For xG, let x¯=xF(G)

Proof of Theorem 1.1, I: first steps

In this section, we focus on the case where G and V have the same underlying field, Fq. We reduce the proof of Theorem 1.1 to a finite list of cases given in Table 4.1, Table 4.2. Here, as well as in future sections, we will consider the realisation of absolutely irreducible FqSLn(q)-modules V(λ) up to quasiequivalence, i.e., up to duality and images under field automorphisms of SLn(q). We begin with the proof of Theorem 1.1 for the small linear groups PSL2(4)PSL2(5), PSL2(9) and PSL4(2),

Proof of Theorem 1.1, II: p-restricted modules

The main result of this section is as follows.

Theorem 5.1

Let V=Vd(q) be a d-dimensional vector space over Fq with q=pe, and let GΓL(V) be almost quasisimple with E(G)/Z(E(G))PSLl+1(q). Further suppose that the restriction of V=V(λ) to E(G) is an absolutely irreducible module of p-restricted highest weight λ. Then either G has a regular orbit on V, or n=l+1, b(G) and λ (up to quasiequivalence) appear in Table 1.1.

By Proposition 4.2, Proposition 4.3, the proof of Theorem 5.1 reduces to an analysis of

Proof of Theorem 1.1, III: tensor product modules

In this section, we aim to prove the following result which will complete the proof of Theorem 1.1 for G,V having the same underlying field Fq.

Theorem 6.1

Let V=V(ι1)V(ι2)(pa) be a highest weight module defined over Fq, q=pe, with ι1, ι2 in Table 4.2 and 1a<e. Let GΓL(V) be almost quasisimple with E(G)/Z(E(G))PSLl+1(q) such that the restriction of V to E(G) is absolutely irreducible. Then either G has a regular orbit on V, or ι1=λ1 and ι2=λ1 or λl up to quasiequivalence. In the latter case, b(G)=2

Absolutely irreducible representations over subfields

In this section, we consider the following embedding of SLn(qc) in SLnc(q) for c1. Let W be the n-dimensional natural module for SLn(qc) over Fqc with standard basis {e1,,en}, and letV=WW(q)W(q2)W(qc1), so that dimV=nc and B={ei1ei2eic|1ijn} is a basis of V.

The semilinear map φ on V sending νei1ei2eic to νqeicei1eic1 for νFqc induces an automorphism of SLn(qc) defined by φ:(aij)(aijq). As discussed in the proof of Proposition 2.4, φ fixes eieiei for 1ic and has

Absolutely irreducible representations over extension fields

In this final section we complete the proof of Theorem 1.1 by considering absolutely irreducible SLn(q)-modules V defined over a field F that can be realised over a proper subfield of F. We have already analysed such modules V over Fq in Sections 5 and 6, so we also assume that k1 (with k as in Theorem 1.1) in this section. The main result of this section is as follows.

Theorem 8.1

Let V=Vd(qk) be a d-dimensional vector space over Fqk with k1, and let GΓL(V) be almost quasisimple with E(G)/Z(E(G))PSLn(q)

Acknowledgements

This paper comprises part of the author's PhD under the supervision of Professor Martin Liebeck. I thank Professor Liebeck for his guidance and careful reading of the paper, the referees for suggesting a number of improvements, and my doctoral examiners Dr. Timothy Burness and Professor David Evans for the corrections they provided to the original version of this work. Financial support from an EPSRC International Doctoral Scholarship at Imperial College London is also gratefully acknowledged.

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