On finite $PσT$-groups

Abstract: Let $\sigma ={\sigma_{i} | i\in I}$ be some partition of the set of all primes $\Bbb{P}$ 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 ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\cal H}$ contains exact one Hall $\sigma_{i}$-subgroup of $G$ for every $i \in I$ such that $\sigma_{i}\cap \pi (G)\ne \emptyset$. 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 $\cal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \cal H$. We obtain a characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-quasinormality is a transitive relation in $G$.