Permutability of minimal subgroups and \(p\)-nilpotency of finite groups. (Q1424106)

Let \(G\) be a finite group and \(G^{\mathcal N}\) be the nilpotent residual of \(G\). In this paper, the authors prove the following main theorem: Let \(p\) be a prime dividing the order of \(G\) with \((|G|,p-1)=1\) and let \(P\) be a Sylow \(p\)-subgroup of \(G\). If every subgroup of \(P\cap G^{\mathcal N}\) with order \(p\) is permutable in \(N_G(P)\) and, when \(p=2\), either every cyclic subgroup of \(P\cap G^{\mathcal N}\) with order 4 is permutable in \(N_G(P)\) or \(P\) is quaternion-free, then \(G\) is \(p\)-nilpotent. Some applications of this theorem are given.