On nearly \({S\Phi}\)-normal subgroups of finite groups

Abstract

Let \(G\) be a finite group, \(H\) a subgroup of \(G\) and \(H_{sG}\) the subgroup of \(H\) generated by all those subgroups of \(H\) which are \(s\)-permutable in \(G\). Then we say that \(H\) is \(\textit{nearly \(S\Phi\)-normal}\) in \(G\) if \(G\) has a normal subgroup \(T\) such that \(HT\unlhd G\) and \(H\cap T\leq \Phi (H)H_{sG}\). In this paper, we study the structure of group \(G\) under the condition that some given subgroups of \(G\) are nearly \(S\Phi\)-normal in \(G\). Some known results are generalised.