A note on extremely primitive affine groups

Let $G$ be a finite primitive permutation group on a set $\Omega$ with nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$. In earlier work, Mann, Praeger and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall's conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of affine groups in order to complete the classification of the extremely primitive groups. In this note, we prove that none of these candidates are extremely primitive, as conjectured by Mann et al.


Introduction
Let G Sym(Ω) be a finite primitive permutation group with point stabilizer H = G α = 1. We say that G is extremely primitive if H acts primitively on each of its orbits in Ω \ {α}. For example, the natural actions of Sym n and PGL 2 (q) of degree n and q + 1, respectively, are extremely primitive. The study of these groups can be traced back to work of Manning [19] in the 1920s and they have been the subject of several papers in recent years [6,7,8,18].
A key theorem of Mann, Praeger and Seress [18,Theorem 1.1] states that every extremely primitive group is either almost simple or affine, and in the same paper they classify all the affine examples up to the possibility of finitely many exceptions. In later work, Burness, Praeger and Seress [6,7] determined all the almost simple extremely primitive groups with socle an alternating, classical or sporadic group. The classification for almost simple groups has very recently been completed in [8], where the remaining exceptional groups of Lie type are handled. We refer the reader to [8,Theorem 3] for the list of known extremely primitive groups.
It is conjectured that the list of extremely primitive affine groups presented in [18] is complete, so [8,Theorem 3] gives a full classification. To describe the current state of play in more detail, let G = V :H be a finite primitive group of affine type, where V = F d p and p is a prime. In [18], the classification is reduced to the case where p = 2 and H is almost simple (that is, H has a unique minimal normal subgroup H 0 , which is nonabelian and simple). A basic tool in the analysis of these groups is [18,Lemma 4.1], which states that G is not extremely primitive if |M(H)| < 2 d/2 , where M(H) is the set of maximal subgroups of H. If H is a sufficiently large almost simple group, then a theorem of Liebeck and Shalev [16] gives |M(H)| < |H| 8/5 and by playing this off against known bounds on the dimensions of irreducible modules for almost simple groups, Mann, Praeger and Seress prove that their list of extremely primitive affine groups is complete up to at most finitely many exceptions.
If ones assumes that the stronger bound |M(H)| < |H| holds, as predicted by a well known (but still open) conjecture of G.E. Wall, then [18,Theorem 4.8] states that the classification of the extremely primitive affine groups (and therefore all extremely primitive  (9) Weil representation  Table 2] groups, given [6,7,8]) is complete up to determining the status of the groups recorded in Table 1. With the exception of the case in the first row, H 0 is a simple group of Lie type over F 2 and V = L(λ) is a 2-restricted irreducible module for H 0 with highest weight λ.
In the table, we express λ in terms of a set of fundamental dominant weights λ 1 , . . . , λ r for H 0 , where r is the untwisted Lie rank of H 0 and the weights are labelled in the usual way (see [4]). Notice that the highest weights in the table are listed up to graph automorphisms of H 0 . Our main result is the following, which shows that none of the candidates in [18, Table  2] are extremely primitive, as conjectured by Mann, Praeger and Seress in [18]. Theorem 1. Let G = V :H be a primitive permutation group of affine type as in Table 1, where V = F d 2 and H is almost simple with socle H 0 . Then G is not extremely primitive.

Preliminaries
Let G = V :H be a primitive permutation group of affine type as in Table 1, where V = The following result is [ Table 1. A complete classification of the maximal subgroups of E 8 (2) up to conjugacy is currently out of reach, but we can calculate |M(H)| in all the remaining cases. We will need the following result in the proof of Theorem 1.  Table 2.
Proof. In each case we use Magma [3] to construct a set of representatives of the conjugacy classes of maximal subgroups of H. Typically, we do this by using the command AutomorphismGroupSimpleGroup to construct Aut(H 0 ) as a permutation group and we then identify H as a subgroup of Aut(H 0 ) (in every case, H is either H 0 , Aut(H 0 ) or a maximal subgroup of Aut(H 0 )). We then use MaximalSubgroups to construct representatives of the classes of maximal subgroups of H and we compute |M(H)| by summing the indices |H : N H (M )| for each representative M . The number α(H 0 ) presented in the table is the maximum value of |M(H)| as we range over all the almost simple groups H with socle H 0 .
For H 0 = Sp 16 (2) and Ω + 16 (2) the command MaximalSubgroups is ineffective and so a slightly modified approach is required. The basic method is identical, but in these cases we use ClassicalMaximals to construct a set of representatives of the classes of maximal subgroups of H, combined with LMGIndex to compute the indices.
We will also need to compute |M(H)| for H = Ω + 18 (2). By Aschbacher's theorem [1] on the subgroup structure of the finite classical groups, each maximal subgroup M of H is either geometric, in which case the possibilities for M are determined up to conjugacy in [12], or M is non-geometric, which means that M is an irreducibly embedded almost simple subgroup.  (2), where Lie (2) is the set of simple groups of Lie type in characteristic 2.
First assume M 0 ∈ Lie(2). By inspecting Lübeck [17], we quickly deduce that the only possibilities for M 0 are L 18 (2), Sp 18 (2) and Ω ± 18 (2) (indeed, these are the only simple groups of Lie type is even characteristic with an 18-dimensional absolutely irreducible representation over F 2 ). Plainly, none of these possibilities can arise. Now assume M 0 ∈ Lie(2). Here we turn to the work of Hiss and Malle [10], which records all the absolutely irreducible representations of finite quasisimple groups up to dimension 250, excluding representations in the defining characteristic. In addition, information on the corresponding Frobenius-Schur indicators and fields of definition is also provided. By inspecting [10], we see that the only possibilities for M 0 are the alternating groups Alt 19 and Alt 20 (note that L 2 (19) does have an 18-dimensional absolutely irreducible representation in even characteristic with indicator +1, but this is defined over F 4 , rather than F 2 ). For M 0 = Alt 19 and Alt 20 , the relevant representation is afforded by the fully deleted permutation module over F 2 . However, we have Alt 19 < Alt 20 < Ω − 18 (2) (see [12, p.187], for example) and so this representation does not embed M 0 in H. (2), then |M| = 115583493125204258236922964476027. Proof. In view of Proposition 2.4, this is an entirely straightforward computation using the ClassicalMaximals command in Magma [3], which returns a set of conjugacy class representatives of the geometric maximal subgroups of H.

Proof of Theorem 1: H classical
We begin the proof of Theorem 1 by handling the groups where H is a classical group. We first observe that several cases can be immediately eliminated by combining Corollary 2.2 with Proposition 2.3.  Table 1 such that H 0 is one of  (2) or Ω + 2k+2 (2), 5 k 7 and V = L(λ k ). We will handle each of these cases in turn, referring to the labels (a), (b) and (c). Proof. In each of these cases, we use Magma to construct the module V and a set of representatives for the conjugacy classes of maximal subgroups of H. To construct V , we use the command IrreducibleModulesBurnside, with the optional DimLim parameter equal to 2000. Apart from the case H = Sp 12 (2), we note that H has a unique ddimensional irreducible module over F 2 , up to graph automorphisms. The group H = Sp 12 (2) has two 64-dimensional irreducible modules, namely L(λ 2 ) and the spin module L(λ 6 ), and they can be distinguished by considering their restrictions to a subgroup Ω + 12 (2) of H. Indeed, the restriction of L(λ 2 ) is irreducible, while the restriction of L(λ 6 ) is reducible (the composition factors are the two 32-dimensional spin modules for Ω + 12 (2)).
Then for each maximal subgroup M of H, we compute the 1-eigenspace C V (x), where x runs through a set of generators X for M . Since fix(M ) = x∈X C V (x), this allows us to compute f (H) precisely (see (1) In each case, it is now routine to verify the bound f (H) < 2 d − 1. By Lemma 2.1, this implies that G is not extremely primitive. Proof. Here H 0 = L k (2) and V = L(λ 3 ) = Λ 3 W , where 7 k 13 and W is the natural module for H 0 . Since the highest weight λ 3 is not invariant under a graph automorphism of H 0 , it follows that H = L k (2). The cases with 7 k 10 can be handled by proceeding as in the proof of Proposition 3.2 and we find that f (L 7 (2)) = 11811, f (L 8 (2)) = 97155, f (L 9 (2)) = 18202348610724300355, f (L 10 (2)) = 413104411638650042899395. However, we will give a uniform argument for all 8 k 13.
With the aid of Magma, it is easy to verify that each maximal subgroup M of H contains an element of order r ∈ {7, 11, 13}. Since V is simply the wedge-cube of W , it is straightforward to calculate dim C V (x) for each element x ∈ H of order r.
For example, suppose H = L 8 (2) and x ∈ H has order 7. Now H has three conjugacy classes of elements of order 7, with representatives where ω ∈ F 2 3 is a primitive 7-th root of unity (see [5,Section 3.2], for example). Notice that we are viewing the conjugacy class representatives as diagonal matrices in SL 8 (8), which is convenient for computing their 1-eigenspaces on V . By considering the eigenvalues of x i on W , it is easy to show that dim C V (x 1 ) = dim C V (x 2 ) = 5 3 + 1 = 11 and dim C V (x 3 ) = 8.
In this way, we find that dim C V (x)  LetH = C 7 be the ambient simple algebraic group over the algebraic closureF 2 and fix a Steinberg endomorphism ofH such thatH σ = H. We may choose σ so thatL σ = L, whereL is a maximal rank subgroup ofH with connected componentL 0 = A 7 1 . Now the restriction of the spin module forH toL 0 is the tensor product of the natural modules for the A 1 factors. In particular, the restriction is irreducible and we deduce that L acts irreducibly on V . Therefore, fix(M ) = 0 for all M ∈ M 1 and thus since the trivial bound dim fix(M ) 2 64 holds for all M ∈ M 2 . By applying Lemma 2.1, we conclude that G is not extremely primitive.
For the remainder of the proof, we may assume that H = Ω + 2k+2 (2) with 5 k 7. The case k = 5 can be handled as in Proposition 3.2 (in order to construct V using the command IrreducibleModulesBurnside, we need to set DimLim equal to 10000) and we get f (H) = 1240917975 < 2 32 − 1.
Next assume k = 6, so H = Ω + 14 (2). Let W be the natural module for H and let M 1 be the set of maximal subgroups of H of one of the following types: There are two conjugacy classes of elements of order 7 in H and both classes have representatives in a reducible subgroup K = Sp 12 (2). In order to compute the 1-eigenspaces of these elements on V , it is convenient to work in the corresponding algebraic groups over F 2 , so writeH = D 7 andK = B 6 . Then the two classes of order 7 in H are represented by the elements , ω, ω 2 , ω 3 , ω 4 , ω 5 , ω 6 ], x 2 = [ωI 2 , ω 2 I 2 , ω 3 I 2 , ω 4 I 2 , ω 5 I 2 , ω 6 I 2 ] inK, where ω ∈F 2 is a primitive 7-th root of unity. LetV = V ⊗F 2 be the spin module forH, which remains irreducible on restriction to a maximal rank subgroupJ = A 6 1 of K; the restriction is the tensor product of the natural 2-dimensional modules for the A 1 factors ofJ. This allows us to compute dim C V (x i ) very easily. For example, the action of x 1 onV is given by and thus dim C V (x 1 ) = 16. Similarly, dim C V (x 2 ) = 10.
Finally, let us assume k = 7, so H = Ω + 16 (2). Let W be the natural module for H. We As in the previous case, H has two conjugacy classes of elements of order 7, represented by elements x 1 , x 2 in a reducible subgroup Sp 14 (2), where dim C W (x 1 ) = 10 and dim C W (x 2 ) = 4. By arguing as in the previous case, we calculate that dim C V (x 1 ) = 32 and dim C V (x 2 ) = 20. Similarly, there is an element x 3 ∈ Sp 14 (2) of order 5 with C W (x 3 ) = 4 and dim C V (x 3 ) = 24. By constructing representatives of the maximal subgroups of H in Magma (using ClassicalMaximals), it is straightforward to check that each M ∈ M(H) contains an element conjugate to either x 1 , x 2 or x 3 (indeed, the order of every maximal subgroup of H is divisible by 7, apart from the subgroups of type O − 4 (2) ≀ Sym 4 ). This justifies the claim and the proof of the proposition is complete.

Proof of Theorem 1: H exceptional
In this final section we complete the proof of Theorem 1 by handling the remaining cases in Table 1 with H 0 an exceptional group of Lie type.  Table 1 with H 0 an exceptional group of Lie type. Then G is not extremely primitive.
Proof. In each case we will demonstrate the existence of a nonzero vector v ∈ V such that the point stabilizer C H (v) is a non-maximal subgroup of H.
In the final three cases, V is the unique nontrivial composition factor of the adjoint module for H 0 (note that the adjoint module is irreducible for H 0 = E ε 6 (2) and E 8 (2), but there are two composition factors when H 0 = E 7 (2)). Write H 0 = (H σ ) ′ , whereH is a simple algebraic group of adjoint type overF 2 and σ is an appropriate Steinberg endomorphism ofH. Let L(H) be the adjoint module forH, which is simply the Lie algebra ofH equipped with the adjoint action ofH, and note that we may view V as a subset of L(H). Recall that the orbits for the action ofH on the set of nilpotent elements of L(H) are called nilpotent orbits.
IfH = E 7 , then the adjoint module L(H) has a unique nontrivial composition factorV and it will be important to note that every nilpotent orbit ofH has a representative inV . Since we are working in even characteristic, we may assume thatH is simply connected and we see that L(H) has a 1-dimensional centre, which is generated by a semisimple element. Therefore the 132-dimensional quotientV contains a representative of every nilpotent orbit as claimed.
It follows from [11, Section 1] that every nilpotent orbit on L(H) has a representative defined over the prime field F 2 . Therefore, in every case we may choose v ∈ V to be a representative of the nilpotent orbit labelled A 2 1 in [15, Tables 22.1.1-22.1.3], which also gives the structure of the stabilizer CH (v). Moreover, CH(v) is σ-stable because it is the only stabilizer of a nilpotent element with its given dimension. Therefore, CH σ (v) = (CH(v)) σ .
First assumeH = E 7 or E 8 . Here H =H σ , so C H (v) = (CH(v)) σ and by inspecting [15] we see that where U i denotes a connected unipotent algebraic group of dimension i. In particular, CH(v) is a proper subgroup of a σ-stable maximal parabolic subgroup ofH, whence C H (v) is non-maximal in H.