A note on \(\mathcal{M}\)-normal embedded subgroups of finite groups (Q6539993)

A non-trivial subgroup \(H\) of a group \(G\) is said to be \(\mathcal{M}\)-normal supplemented in \(G\), if there exists a normal subgroup \(B\) of \(G\) such that \(G=HB\) and \(H_{1}B < G\) for every maximal subgroup \(H_{1}\) of \(H\). The subgroup \(H\) is said to be \(\mathcal{M}\)-normal embedded in \(G\), if there exists a normal subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K=1\) or \(H \cap K\) is \(\mathcal{M}\)-normal supplemented in \(G\).\N\NThe main result of this note is Theorem 1.5: Let \(p\) be a prime dividing the order of a group \(G\), \(P \in \mathrm{Syl}_{p}(G)\) and \(N \trianglelefteq G\). Suppose that \(P\) has a nontrivial subgroup \(D\) such that, for every normal subgroup \(H\) of \(P\) with \(|H|=|D|\), \(H \cap N \leq \Phi(H)\). Then \(N\) is \(p\)-nilpotent. \N\NThis theorem generalizes (and simplifies the proofs) of the results obtained by \textit{R. Chen} et al. in [Math. Scand. 127, No. 2, 243--251 (2021; Zbl 07465733)].