On \(\mathcal M\)-permutable Sylow subgroups of finite groups.
Let \(G\) be a finite group and \(p\) a prime divisor of \(|G|\). A nontrivial \(p\)-subgroup \(P\) of \(G\) is called \(\mathcal M\)-permutable in \(G\) if there exists a set \(\mathcal M_d(P)=\{P_1,\dots,P_d\}\) of maximal subgroups \(P_i\) of \(P\) and a subgroup \(B\) of \(G\) such that \[ \bigcap^d_{i=1}P_i=\Phi(P)\quad\text{and}\quad|P:\Phi(P)|=p^d\tag{1} \] and \[ G=PB\quad\text{and}\quad P_iB=BP_i<G\quad\text{for any }P_i\text{ of }\mathcal M_d(P).\tag{2} \] If \(p\) is an odd prime divisor of the order of \(G\) and \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(G\) is \(p\)-nilpotent if and only if \(N_G(P)\) is \(p\)-nilpotent and \(P\) is \(\mathcal M\)-permutable in \(G\) (Theorem 3.1). If \(\mathcal F\) is a saturated formation containing all supersoluble groups and \(H\) is a normal subgroup of \(G\) such that \(G/H\in\mathcal F\) and every noncyclic Sylow subgroup of \(H\), \(F(H)\) or \(F^*(H)\) is \(\mathcal M\)-permutable in \(G\), then \(G\in\mathcal F\) (Theorem 3.2, 3.4 and 3.6., respectively).
scientific article; zbMATH DE number 5824936
scientific article