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Register a new userOn a lattice characterization of finite soluble $PST$-groups
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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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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 $sigma ={sigma_{i} | iin I}$ be some partition of the set of all primes $Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $sigma $-full if $G$ has a Hall $sigma _{i}$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $sigma$-permutable in $G$ provided $G$ is $sigma $-full and $A$ permutes with all Hall $sigma _{i}$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $sigma$-permutability is a transitive relation in $G$, that is, if $K$ is a ${sigma}$-permutable subgroup of $H$ and $H$ is a ${sigma}$-permutable subgroup of $G$, then $K$ is a ${sigma}$-permutable subgroup of $G$.
Let ${frak F}$ be a class of group and $G$ a finite group. Then a set $Sigma $ of subgroups of $G$ is called a emph{$G$-covering subgroup system} for the class ${frak F}$ if $Gin {frak F}$ whenever $Sigma subseteq {frak F}$. We prove that: {sl If a set of subgroups $Sigma$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $Sigma$ is a $G$-covering subgroup system for the classes of all $sigma$-soluble and all $sigma$-nilpotent groups, and for the class of all $sigma$-soluble $Psigma T$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook.
Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $delta_k$-values $a,bin G$ of coprime orders. In the course of the proof we establish the following result of independent interest: Let $P$ be a Sylow $p$-subgroup of $G$. Then $Pcap G^{(k)}$ is generated by $delta_k$-values contained in $P$. This is related to the so-called Focal Subgroup Theorem.
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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