On the Saxl graph of a permutation group

Abstract

Let G be a permutation group on a set Ω.

Saxl graphs of primitive affine groups with sporadic point stabilisers

“…In addition, Chen and Huang [8] have calculated the valency of Σ(G) when G is an almost simple primitive group with socle an alternating group and soluble point stabiliser. (As noted in [6,8], and in our Lemma 2.2, Σ(G) has the property that every two vertices have a common neighbour if its valency is at least |Ω|/2. )…”

“…(5, 63), (9, 9), (10,6), (11,5), (16,3), (24, 2) M 12 (10,9), (11,9), (15,5), (16,4), (20, 3), (22, 2), (29, 2), (30, 2), (32, 2), (34, 2) 2.M 12 (6, 66), (10,9), (12,8) 4, it remains to consider the faithful irreducible F r G-modules with r R, for each pair (d, R) given in the second column. As explained in the proof of the proposition, R is simply the largest integer such that f (G, d, R) < 1/2, so several values of r R are readily ruled out either because (i) they are not prime powers, (ii) the d-dimensional representation in question is not actually realised over F r (as one may verify using either the relevant character table in GAP [4,12] or by referring to Hiss and Malle [15]), or (iii) the representation is realised but b(GV ) = 2, i.e.…”

“…Saxl graphs are named in honour of Jan Saxl, who initiated a project to classify the finite primitive permutation groups with base size 2. They were introduced by Burness and Giudici [6], who made the following conjecture.…”

“…Conjecture 1.1 (Burness and Giudici [6]). If G is a finite primitive permutation group with b(G) = 2, then every two vertices of Σ(G) have a common neighbour.…”

On the Saxl graphs of primitive groups with soluble stabilisers

“…Let G Sym(Ω) be a base-two finite transitive permutation group with point stabiliser H. In [10], Burness and Giudici define the Saxl graph Σ(G) of G as follows: the vertex set is Ω and two vertices are joined by an edge if and only if they form a base for G. Clearly, Σ(G) is vertex-transitive and it is easy to see that Σ(G) is connected if G is primitive (as discussed in [10], if G is imprimitive then Σ(G) can have arbitrarily many connected components). In addition, Σ(G) is a complete graph if and only if G is a Frobenius group.…”