Affiliation:
1. School of Mathematical Sciences , Xiamen University , Xiamen 361005 , P. R. China
2. School of Mathematics , University of Bristol , Bristol BS8 1UG , United Kingdom
Abstract
Abstract
Let 𝐺 be a permutation group on a set Ω, and recall that a base for 𝐺 is a subset of Ω such that its pointwise stabiliser is trivial.
In a recent paper, Burness and Giudici introduced the Saxl graph of 𝐺, denoted
Σ
(
G
)
\Sigma(G)
, with vertex set Ω and two vertices adjacent if and only if they form a base for 𝐺.
If 𝐺 is transitive, then
Σ
(
G
)
\Sigma(G)
is vertex-transitive, and it is natural to consider its valency (which we refer to as the valency of 𝐺).
In this paper, we present a general method for computing the valency of any finite transitive group, and we use it to calculate the exact valency of every primitive group with stabiliser a Frobenius group with cyclic kernel.
As an application, we calculate the valency of every almost simple primitive group with an alternating socle and soluble stabiliser, and we use this to extend results of Burness and Giudici on almost simple primitive groups with prime-power or odd valency.
Subject
Algebra and Number Theory
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to appear in Algebra Number Theory.
Cited by
4 articles.
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