The Classification of Extremely Primitive Groups

Abstract

Dedicated to the memory of Jan Saxl 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 $\Omega \setminus \{\alpha \}$. These groups arise naturally in several different contexts, and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger, and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger, and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.

A note on extremely primitive affine groups

“…Let G Sym(Ω) be a finite extremely primitive group with point stabiliser H. By [24, Theorem 1.1], G is either almost simple or an affine type group. The main results in [7,8,10] provide a classification of the almost simple extremely primitive groups (see [10,Table 1] for the complete list). Therefore, we may assume G = V :H AGL(V ) is an affine group, where V = F d p for some prime p and H GL(V ) is irreducible.…”

“…For instance, they feature in the original constructions of some of the sporadic simple groups (in particular J 2 and HS) and they arise in the study of permutation groups with restricted movement (see [28], for example). As far back as the 1920s, one can find important work of Manning [25] on extremely primitive groups and they have been the main subject of several papers in recent years [7,8,9,10,24].…”

On the Saxl graphs of primitive groups with soluble stabilisers

“…Proof. The possibilities for H are recorded in [30, Table 5.2] and [17,Proposition 4.2] states that b(G) = 2 whenever H is the normaliser of a maximal torus (soluble or otherwise). We proceed by carefully inspecting the proof of [17,Proposition 4.2] in the relevant cases with H soluble.…”

“…If G 0 = E 8 (q) then one checks that the bound on Q(G) in the proof of [17,Lemma 4.3] is sufficient and we note that H is insoluble when…”

“…These groups arise naturally in many applications of bases to other problems. For example, see [17] on the classification of extremely primitive groups and [14] for applications concerning the 2-generation properties of almost simple groups. Although a complete classification of the base-two primitive groups remains out of reach, there has been some significant progress.…”