A note on extremely primitive affine groups
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Archiv der Mathematik
A note on extremely primitive affine groups Timothy C. Burness and Adam R. Thomas
Abstract. Let G be a finite primitive permutation group on a set Ω with non-trivial point stabilizer Gα . We say that G is extremely primitive if Gα acts primitively on each of its orbits in Ω\{α}. 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. Mann et al. have conjectured that none of these affine candidates are extremely primitive and our main result confirms this conjecture. Mathematics Subject Classification. 20B15. Keywords. Primitive groups, Affine groups, Maximal subgroups.
1. Introduction. Let G Sym(Ω) be a finite primitive permutation group with point stabilizer H = Gα = 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 Symn and PGL2 (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 [6–8,18]. 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
T.C. Burness and A.R. Thomas
Arch. Math.
very recently been completed in [8], where the remaining exceptional groups of Lie type are handled. We refer the reader to [8, Theorem 4] 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 4] 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 = Fdp 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 H0 , which is non-abelian 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)| < 2d/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 again
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