On finite $P\sigma T$-groups

Abstract

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$\sigma$-soluble} if every chief factor $H/K$ of $G$ is a ${\sigma }_i$-group for some $i=i(H/K)$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma$-set} of $G$ if every member $\ne 1$ of $\mathcal{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i\in \sigma$ and $\mathcal{H}$ contains exact one Hall ${\sigma }_i$-subgroup of $G$ for every $i\in I$ such that ${\sigma }_i\cap \pi (G)\ne \varnothing$. A subgroup $A$ of $G$ is said to be \emph{$\sigma$-permutable} or \emph{$\sigma$-quasinormal} in $G$ if $G$ has a complete Hall $\sigma$-set $\mathcal{H}$ such that $A{H}^x=H^{x}A$ for all $x\in G$ and all $H\in \mathcal{H}$. We obtain a characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-quasinormality is a transitive relation in $G$.

arXiv comment11 pages