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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.
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
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 $sigma$-
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
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 establi
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 st