A note on m-embedded subgroups of finite groups (Q2813418)

Let \(G\) be a finite group and the odd prime \(p\) a divisor of \(|G|\). The subgroup \(A\) is called m-embedded in \(G\) if \(G\) has a subnormal subgroup \(T\) and a \(\{1\leq G\}\)-embedded subgroup \(C\) such that \(G=AT\) and \(T\cap A\leq C\leq A\). This and other similar structure properties of finite groups were studied by \textit{W. Guo} and \textit{A. N. Skiba} [Sci. China, Math. 54, No. 9, 1909--1926 (2011; Zbl 1255.20018)], some of the results are extended by the authors of the present paper. Namely, let \(P\) be a \(p\)-Sylow subgroup of \(G\). If every maximal subgroup \(P_1\) of \(P\) is m-embedded in \(G\) and either the normalizers of \(P\) or of each maximal subgroup \(P_1\) of \(P\) are \(p\)-nilpotent then \(G\) is \(p\)-nilpotent. Furthermore, let \(G\) be \(p\)-soluble. If either each maximal subgroup \(P_1\) of \(P\) is m-embedded in \(G\), or each maximal subgroup of \(F_p(G)\) containing \(O_{p'}(G)\) is m-embedded in \(G\) then \(G\) is \(p\)-supersoluble.