Base sizes for primitive groups with soluble stabilisers

Abstract

Let G be a finite primitive permutation group on a set Ω with point stabiliser H. Recall that a subset of Ω is a base for G if its pointwise stabiliser is trivial. We define the base size of G, denoted b(G, H), to be the minimal size of a base for G. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that b(G, H) 4 if G is soluble. In this paper we extend Seress' result by proving that b(G, H) 5 for all finite primitive groups G with a soluble point stabiliser H. This bound is best possible. We also determine the exact base size for all almost simple groups and we study random bases in this setting. For example, we prove that the probability that 4 random elements in Ω form a base tends to 1 as |G| tends to infinity.

On Valency Problems of Saxl Graphs

“…Proof. This is given in the proof of [5,Lemma 4.7]. Here we give another proof using the strategy introduced in Section 3.…”

“…Our first application concerns the almost simple primitive groups with socle an alternating group. Let G be an almost simple primitive group with socpGq " A n and soluble stabiliser H. Note that valpG, Hq " 0 if bpGq ą 2, and those groups with bpGq " 2 are classified in [5]. In the following theorem, µ denotes the Möbius function and φ denotes the Euler totient function.…”

“…Let G be an almost simple primitive group with stabiliser H. If bpGq " 2, socpGq " A n and H is soluble, then by [5,19], pG, Hq is exactly one given in Table 1. Our aim is to calculate valpG, Hq listed in Table 1.…”

“…If G is a soluble primitive group, then a theorem of Seress [23] shows that bpGq 4 and this has very recently been extended by Burness [5], who has established the bound bpGq 5 for any finite primitive group G with a soluble point stabiliser (in both cases, the bounds are best possible). In addition, [5,Theorem 2] gives the exact base size for every almost simple primitive group with a soluble stabiliser.…”

“…This conjecture was proved by Liebeck and Shalev [20] using probabilistic methods and it is now known that c " 7 is the optimal constant (in fact, bpGq " 7 if and only if G is the Mathieu group M 24 acting on 24 points); see the sequence of papers [3,8,10,11] by Burness et al Furthermore, almost simple primitive groups with bpGq " 6 have been determined in [4, Theorem 1]. If G is a soluble primitive group, then a theorem of Seress [23] shows that bpGq 4 and this has very recently been extended by Burness [5], who has established the bound bpGq 5 for any finite primitive group G with a soluble point stabiliser (in both cases, the bounds are best possible). In addition, [5,Theorem 2] gives the exact base size for every almost simple primitive group with a soluble stabiliser.…”