Regular orbits of quasisimple linear groups II
Introduction
A permutation group is said to have a regular orbit on Ω if there exists 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 . For instance, if V is a finite-dimensional vector space over a finite field, , and all orbits of G on 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 -conjecture [14], asserting that the number of conjugacy classes of GV, with coprime to , is at most . 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 -conjecture. A classification of pairs where G has a regular orbit on V and was completed by Köhler and Pahlings [26], building on work of Goodwin [15], [16] and Liebeck [30].
A subset 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 . For example, if G has a regular orbit on V, then . Each element is characterised by its action on a base, so . In recent years, there has been a number of advancements towards classifying finite primitive groups H with small bases. A primitive group with is cyclic, so the smallest case of interest is where . 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 and [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 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 of G, and the quotient of G by its Fitting subgroup is almost simple. The Fitting subgroup of an almost quasisimple group is generated by the group of scalars contained in G, as well as the set of field automorphisms in G that commute with . A primitive affine group has base size 2 if and only if 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 . In this paper, we investigate the case where G is a member of the class of Aschbacher's theorem. Slightly more broadly than this, we consider almost quasisimple such that acts absolutely irreducibly on V.
The pairs where G has a regular orbit on V have been classified for a sporadic or alternating group [10], [11], and for G of Lie type with in the aforementioned proof of the -conjecture. The authors of [29] showed that for with quasisimple, as long as V is not the natural module for . Moreover, Guralnick and Lawther [18] classified where G is a simple algebraic group over an algebraically closed field of characteristic 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 , where is almost quasisimple and is a group of Lie type that acts absolutely irreducibly on V, with . The first paper [28] dealt with the cross-characteristic representations of G, while the present paper considers the case where and V is an absolutely irreducible module for in defining characteristic. The final paper in the series will consider the remaining almost quasisimple groups of Lie type in defining characteristic.
We say that two representations of (and their corresponding modules) are quasiequivalent if there exists such that is equivalent to . For an almost quasisimple group of Lie type , and a natural number i, we define if , and otherwise.
The main theorem of this work is as follows. Theorem 1.1 Let be a d-dimensional vector space over with , and let be an almost quasisimple group such that with and . Suppose that the restriction of V to is an absolutely irreducible module of highest weight λ. Set . Either G has a regular orbit on V, or up to quasiequivalence, λ and n appear in Table 1.1 if 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 and , then G has a regular orbit on V. Corollary 1.3 Let G, V be as in Theorem 1.1. Then either V is the natural module for G, or .
We now give some remarks on the content of Table 1.1, Table 1.2. Remark 1.4 The notation 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 have . The rows in Table 1.1, Table 1.2 for are asterisked because if contains no field automorphisms, then , and only if either and G contains scalars in , or G contains field automorphisms. 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 , the parameter k is either an integer , or can be written as , with . The entry for and is asterisked because, by Proposition 5.15, G has a regular orbit on V if G is quasisimple. Otherwise, . The entries for with in Table 1.1, Table 1.2 are asterisked because by Proposition 5.14, Proposition 8.13, , where δ is 1 if G contains a graph automorphism and zero otherwise. The row for 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 . If or , then by Proposition 6.7, there is no regular orbit of G on V unless , or and , where K is the kernel of the action of . 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 . Each row in Table 1.2 with for some describes an -module over a subfield . In each of these cases, . The construction of such modules is described in Section 7. The papers [10], [11], [15], [16], [26] mentioned in the introduction each assume that V (as in Theorem 1.1) is a faithful -dimensional -module, with prime such that acts irreducibly, but not necessarily absolutely irreducibly, on V. We can reconcile this with our study of absolutely irreducible modules for over finite fields of arbitrary prime power order, following [15, §3]. Define , , and . Then is absolutely irreducible, and , where ϕ is a field automorphism of order t.
Corollary 1.5
Let G, V be as in Theorem 1.1, and further suppose G is quasisimple. If , then either G has a regular orbit on V, or 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 -module V, then every vector is fixed by a prime order element in G. For each V, we then set out to give a proof by showing that exceeds the sum of the sizes of the fixed point spaces of prime order elements (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 corresponding to . 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 , a simple algebraic group of type , has no regular orbit on an irreducible -module , but according to Theorem 1.1, the corresponding finite group has a regular orbit on V realised over , where . For example, when , and , then [18, Proposition 3.1.8] asserts that there is no regular orbit under the action of , but we prove in Proposition 6.7 that there is a regular orbit on V under the action of when 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 where are dealt with in Section 5 for λ p-restricted and Section 6 otherwise. Section 7 deals with absolutely irreducible modules with that are not realised over a proper subfield of . 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 be a simple algebraic group over , p prime. Let be a fixed maximal torus of , and Φ be the root system of with respect to T, with base of simple roots, and corresponding fundamental dominant weights . We define a partial ordering on weights by saying if and only if is a non-negative linear combination of simple roots. By a theorem of Chevalley, the irreducible modules of 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 denote the set of elements of G of projective prime order s, and denote the set of elements of G with projective prime order coprime to s.
Proposition 3.1 Let be an almost quasisimple group, acting irreducibly on the d-dimensional module over . Set and let be a set of conjugacy class representatives of elements of projective prime order in G. For , let [28, Proposition 3.1]
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, . 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 -modules up to quasiequivalence, i.e., up to duality and images under field automorphisms of . We begin with the proof of Theorem 1.1 for the small linear groups , and ,
Proof of Theorem 1.1, II: p-restricted modules
The main result of this section is as follows. Theorem 5.1 Let be a d-dimensional vector space over with , and let be almost quasisimple with . Further suppose that the restriction of to is an absolutely irreducible module of p-restricted highest weight λ. Then either G has a regular orbit on V, or , and λ (up to quasiequivalence) appear in Table 1.1.
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 having the same underlying field . Theorem 6.1 Let be a highest weight module defined over , , with , in Table 4.2 and . Let be almost quasisimple with such that the restriction of V to is absolutely irreducible. Then either G has a regular orbit on V, or and or up to quasiequivalence. In the latter case,
Absolutely irreducible representations over subfields
In this section, we consider the following embedding of in for . Let W be the n-dimensional natural module for over with standard basis , and let so that and is a basis of V.
The semilinear map φ on V sending to for induces an automorphism of defined by . As discussed in the proof of Proposition 2.4, fixes for and has
Absolutely irreducible representations over extension fields
In this final section we complete the proof of Theorem 1.1 by considering absolutely irreducible -modules V defined over a field that can be realised over a proper subfield of . We have already analysed such modules V over in Sections 5 and 6, so we also assume that (with k as in Theorem 1.1) in this section. The main result of this section is as follows.
Theorem 8.1 Let be a d-dimensional vector space over with , and let be almost quasisimple with
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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