On finite $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$σ$-soluble} if every chief factor $H/K$ of $G$ is a $σ_{i}$-group for some $i=i(H/K)$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $i \in I$ such that $σ_{i}\cap π(G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be \emph{$σ$-permutable} or \emph{$σ$-quasinormal} in $G$ if $G$ has a complete Hall $σ$-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 $σ$-soluble groups $G$ in which $σ$-quasinormality is a transitive relation in $G$.

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