A note on Hall S-permutably embedded subgroups of finite groups

Abstract

Let \(G\) be a finite group. Recall that a  subgroup \(A\) of \(G\) is said to  permute with a subgroup \(B\) if \(AB=BA\). A subgroup \(A\) of \(G\) is said to be \(S\)-quasinormal or \(S\)-permutable in \(G\)   if \(A\) permutes with all Sylow subgroups of \(G\). Recall also that \(H^{s G}\) is the \(S\)-permutable closure of \(H\) in \(G\), that is, the intersection of all such \(S\)-permutable subgroups of \(G\) which contain \(H\). We say that \(H\) is Hall \(S\)-permutably embedded in \(G\) if \(H\) is a Hall subgroup of the \(S\)-permutable closure \( H^{s G} \) of \(H\) in \(G\). We prove that the  following conditions are equivalent: (1) every subgroup of \(G\) is Hall \(S\)-permutably embedded in \(G\); (2) the nilpotent residual  \(G^{\frak{N}}\)  of \(G\) is a Hall cyclic of square-free order subgroup of \(G\); (3) \(G = D \rtimes M\) is a split extension of a cyclic subgroup \(D\)  of square-free order by a nilpotent group \(M\), where \(M\) and \(D\) are both Hall subgroups of \(G\).