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Register a new userOn 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 vertex 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.
A graph is edge-primitive if its automorphism group acts primitively on the edge set, and 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs. In this paper, we present a classification for those edge-primitive graphs which are 2-arc-transitive and have soluble edge-stabilizers.
Let $sigma ={sigma_{i} | iin I}$ be a partition of the set of all primes $Bbb{P}$ and $G$ a finite group. Let $sigma (G)={sigma _{i} : sigma _{i}cap pi (G) e emptyset$. A set ${cal H}$ of subgroups of $G$ is said to be a complete Hall $sigma $-set of $G$ if every member $ e 1$ of ${cal H}$ is a Hall $sigma _{i}$-subgroup of $G$ for some $iin I$ and $cal H$ contains exactly one Hall $sigma _{i}$-subgroup of $G$ for every $i$ such that $sigma _{i}in sigma (G)$. We say that $G$ is $sigma$-full if $G$ possesses a complete Hall $sigma $-set. A complete Hall $sigma $-set $cal H$ of $G$ is said to be a $sigma$-basis of $G$ if every two subgroups $A, B incal H$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $sigma$-basis. In particular, we prove that if $G$ has a a $sigma$-basis, then $G$ is generalized $sigma$-soluble, that is, $G$ has a complete Hall $sigma $-set and for every chief factor $H/K$ of $G$ we have $|sigma (H/K)|leq 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281--309], we prove the following Theorem A. Suppose that $G$ is $sigma$-full. Then every complete Hall $sigma$-set of $G$ forms a $sigma$-basis of $G$ if and only if $G$ is generalized $sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $sigma (H/K)=sigma (G/C_{G}(H/K))$ or $sigma (H/K) ={sigma _{i}}$ and $G/C_{G}(H/K)$ is a $sigma _{i} cup sigma _{j}$-group for some $i e j$.
Let $G$ be a reductive algebraic group over an algebraically closed field and let $V$ be a quasi-projective $G$-variety. We prove that the set of points $vin V$ such that ${rm dim}(G_v)$ is minimal and $G_v$ is reductive is open. We also prove some results on the existence of principal stabilisers in an appropriate sense.
Let $mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${cal L}_{mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}in mathfrak{F}$. A chief factor $H/K$ of $G$ is $mathfrak{F}$-central in $G$ if $(H/K)rtimes (G/C_{G}(H/K)) inmathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $mathfrak{F}$-central in $G$ for every subgroup $Ain {cal L}_{mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}leq Z_{infty}(G/A_{G})$ for every subgroup $Ain {cal L}_{mathfrak{N}}(G)$, where $mathfrak{N}$ is the class of all nilpotent groups.
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