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تسجيل مستخدم جديدOn the Saxl graphs of primitive groups with soluble stabilisers
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Let $G$ be a transitive permutation group on a finite set $Omega$ and recall that a base for $G$ is a subset of $Omega$ with trivial pointwise stabiliser. The base size of $G$, denoted $b(G)$, is the minimal size of a base. If $b(G)=2$ then we can study the Saxl graph $Sigma(G)$ of $G$, which has vertex set $Omega$ and two vertices are adjacent if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most $2$ when $G$ is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of $Sigma(G)$ and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of $Sigma(G)$.
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Let $G$ be a permutation group on a set $Omega$ and recall that a base for $G$ is a subset of $Omega$ such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of $G$, denoted $Sigma(G)$, with ver
tex set $Omega$ and two vertices adjacent if they form a base. If $G$ is transitive, then $Sigma(G)$ 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.
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