A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups

Abstract

Let \(\sigma =\{\sigma_{i} \mid i\in I\}\) be a 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 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 exactly one Hall \(\sigma_{i}\)-subgroup of \(G\) for every \(i\) such that \(\sigma_{i}\cap \pi (G)\ne \varnothing\). A subgroup \(A\) of \(G\) is said to be \({\sigma}\)-quasinormal or \({\sigma}\)-permutable in \(G\) if \(G\) has a complete Hall \(\sigma\)-set \(\mathcal H\) such that \(AH^{x}=H^{x}A\) for all \(x\in G\) and all \(H\in \mathcal H\). We obtain a new characterization of finite \(\sigma\)-soluble groups \(G\) in which \(\sigma\)-permutability is a transitive relation in \(G\).