A note on \(S\)-semipermutable and \(S\)-permutably embedded subgroups of finite groups (Q6601249)

Let \(G\) be a finite group and \(H\) be a subgroup of \(G\). Then \(H\) is called an \(S\)-permutable subgroup of \(G\) if \(H\) permutes with every Sylow subgroup of \(G\) while \(H\) is called an \(S\)-semipermutable subgroup of \(G\) if \(HP=PH\) for every Sylow \(p\)-subgroup of \(G\) and for every prime \(p\) such that \(p\not \in \pi (H)\).\N\NIf for every \(p\in \pi(H)\), a Sylow \(p\)-subgroup of \(H\) is also a Sylow \(p\)-subgroup of some \(S\)-permutable subgroup of \(G\), then \(H\) is said to be \(S\)-permutably embedded in \(G\).\N\NIf \(P\) is a non-trivial finite \(p\)-group and \(d\) is the smallest number of elements needed to generate \(P\), then we denote by \(\mathcal{M}_d(P)\) a set of \(d\) maximal subgroups of \(P\) whose intersection is the Frattini subgroup \(\Phi(P)\).\N\NThe authors characterize finite \(p\)-nilpotent and \(p\)-supersolvable \(G\) using maximal subgroups of a Sylow \(p\)-subgroup \(P\) of \(G\) that belong to some \(\mathcal{M}_d(P)\) and the concepts of \(S\)-semipermutable and\N\(S\)-permutably embedded subgroups.\N\NThey obtain the following results.\N\NTheorem 1.2 Let \(G\) be a \(p\)-solvable group and \(P\) be a Sylow \(p\)-subgroup of \(G\). Then \(G\) is \(p\)-supersolvable if and only if for every element \(P_1\) of some fixed \(\mathcal{M}_d(P)\), the subgroup \(P_1\cap O^p(G^*_p)\) is \(S\)-semipermutable or \(S\)-permutably embedded in \(G\).\N\NTheorem 1.3 Let \(p\) be a prime dividing the order of a group \(G\) and \(P\) be a Sylow \(p\)-subgroup of \(G\). Then \(G\) is \(p\)-nilpotent if and only if \(N_G(P)\) is \(p\)-nilpotent and for every element \(P_1\) of some fixed \(\mathcal{M}_d(P)\), the subgroup \(P_1\cap O^p(G^*_p)\) is \(S\)-semipermutable or \(S\)-permutably embedded in \(G\).\N\NTheorem 1.4 Let \(p\) be a prime dividing the order of a group \(G\) such that \((|G|, p-1)=1\) and \(P\) be a Sylow \(p\)-subgroup of \(G\). Then \(G\) is \(p\)-nilpotent if and only if for every element \(P_1\) of some fixed \(\mathcal{M}_d(P)\), the subgroup \(P_1\cap O^p(G^*_p)=P_1\cap O^p(G)\) is \(S\)-semipermutable or \(S\)-permutably embedded in \(G\).