A note on extremely primitive affine groups

Abstract

Let G be a finite primitive permutation group on a set $$\Omega $$ Ω with non-trivial point stabilizer $$G_{\alpha }$$ G α . We say that G is extremely primitive if $$G_{\alpha }$$ G α acts primitively on each of its orbits in $$\Omega {\setminus } \{\alpha \}$$ 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.

The Classification of Extremely Primitive Groups

“…If x ∈ G is a long (or short) root element, then |x G | > q 16 = b 1 and Corollary 2.13 gives |x G ∩ H| 24(q + 1) 4 = a 1 . As noted above, for all other nontrivial elements we have |x G | > q 22 and we observe that…”

“…indivisible by q 24 . Then since |M | > q 22 , it follows that M is one of the subgroups listed in [17,Lemma 4.23]. But none of these subgroups have order divisible by q 20 , so we have reached a contradiction and we conclude that G is not extremely primitive.…”

“…By adopting a different approach, we prove in [22] that none of the groups recorded in [65, Table 2] are extremely primitive. In particular, this reduces the classification of the affine extremely primitive groups to Wall's conjecture for almost simple groups.…”

“…Let W ( Ḡ) = O + 4 (3) be the Weyl group of Ḡ and note that q is even and G contains graph automorphisms (see [50,Table 5.2]). It will be useful to note that if x ∈ G has prime order, and x is not a long or short root element, then |x G | > q 22 (minimal if x is an involution in the class labelled ( Ã1 ) 2 ).…”

The classification of extremely primitive groups

“…By adopting a different approach, we prove in [20] that none of the groups recorded in [62, Table 2] are extremely primitive. In particular, this reduces the classification of the affine extremely primitive groups to Wall's conjecture for almost simple groups.…”