On <i>IC</i> Φ <i> _S </i> – subgroups of finite groups (Q6594883)

Let \(H\) be a subgroup of a finite group \(G\) and \(H_{sG}\) be the subgroup generated by all subgroups of \(H\) which are \(s\)-permutable that is which permute with every Sylow subgroup of \(G\). Then \(H\) is called an IC\(\Phi_s\)-subgroup of \(G\) if \((H\cap [H,G])H_G/H_G\subseteq \Phi(H/H_G)H_{sG}/H_G\).\N\NIn this paper, the following result is proved.\N\NTheorem 1.5 Let \(G\) be a finite group, \(N\) be a normal subgroup of \(G\), \(p\) be a prime such that \(p\in\pi(N)\) and \(P\) be a Sylow \(p\)-subgroup of \(N\). If there exists a non-trivial proper subgroup \(D\) of \(P\) such that every subgroup of \(P\), of order \(|D|\) and of order \(4\) if \(|D|=2\) and \(P\) is a non abelian \(2\)-group, is an IC\(\Phi_s\)-subgroup of \(G\), then \(N\) is a subgroup of the product \(Z_{p\mathcal{U}}(G)\) of all normal subgroups \(H\) of \(G\) such that every \(p\)-chief factor of \(G\) below \(H\) is cyclic.\N\NThis theorem extends some results of the paper [\textit{Y. Gao} and \textit{X. Li}, Commun. Algebra 50, No. 5, 2139--2148 (2022; Zbl 1505.20011)].