On an application of the lattice of \(\sigma\)-permutable subgroups of a finite group

Abstract

Let \(\sigma =\{\sigma_{i} \mid i\in I\}\) be some partition of the set of all primes and \(G\) a finite group. Then \(G\) is said to be \(\sigma\)-full if \(G\) has a Hall \(\sigma _{i}\)-subgroup for all \(i\); \(\sigma\)-primary if \(G\) is a \(\sigma _{i}\)-group for some \(i\); \(\sigma\)-soluble if every chief factor of \(G\) is \(\sigma\)-primary; \(\sigma\)-nilpotent if \(G\) is the direct product of \(\sigma\)-primary groups; \(G^{\mathfrak{N}_{\sigma}}\) denotes the \(\sigma\)-nilpotent residual of \(G\), that is, the intersection of all normal subgroups \(N\) of \(G\) with \(\sigma\)-nilpotent quotient \(G/N\).

A subgroup \(A\) of \(G\) is said to be: \(\sigma\)-permutable in \(G\) provided \(G\) is \(\sigma\)-full and \(A\) permutes with all Hall \(\sigma _{i}\)-subgroups \(H\) of \(G\) (that is, \(AH=HA\)) for all \(i\); \(\sigma\)-subnormal in \(G\) if there is a subgroup chain \(A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G\) such that either \(A_{i-1} \trianglelefteq A_{i}\) or \(A_{i}/(A_{i-1})_{A_{i}}\) is \(\sigma\)-primary for all \(i=1, \ldots , n\).

Let \(A_{\sigma G}\) be the subgroup of \(A\) generated by all \(\sigma\)-permutable subgroups of \(G\) contained in \(A\) and \(A^{\sigma G}\) be the intersection of all \(\sigma\)-permutable subgroups of \(G\) containing \(A\).

We prove that if \(G\) is a finite \(\sigma\)-soluble group, then the \(\sigma\)-permutability is a transitive relation in \(G\) if and only if \(G^{\mathfrak{N}_{\sigma}}\) avoids the pair \((A^{\sigma G}, A_{\sigma G})\), that is, \(G^{\mathfrak{N}_{\sigma}}\cap A^{\sigma G}= G^{\mathfrak{N}_{\sigma}}\cap A_{\sigma G}\) for every \(\sigma\)-subnormal subgroup \(A\) of \(G\).