The local permutability of primary subgroups (Q6601877)

The authors introduce the concept of \(X\)-\(ss\)-semipermutable subgroups of finite groups. Let \(H\) be a subgroup of a finite group \(G\). If there exists a supplement \(T\) of \(H\) in \(G\), such that there exists some element \(x\in X\), \(HP^{x}=P^{x}H\), where \(\gcd(p, |H|)=1\), \(\emptyset\neq X\subseteq G\) and \(P\in\mathrm{Syl}_{p}(T)\), then we call \(H\) is \(X\)-\(ss\)-semipermutable in \(G\). The authors investigate the structure of a finite group by the \(X\)-\(ss\)-semipermutability of some primary subgroups.