On \(\mathfrak F_{hq}\)-supplemented subgroups of a finite group
Let \(G\) be a finite group, and \(H\) a subgroup of \(G\). Suppose there is a quasinormal subgroup \(N\) of \(G\) such that the subgroup \(H N\) of \(G\) is a Hall subgroup of \(G\), and \((H \cap N) H_{G} / H_{G}\) is contained in \(Z_{\mathfrak F}(G / H_{G})\). Then \(H\) is said to be \({\mathfrak F}_{hq}\)-supplemented in \(G\). Here, a subgroup \(N\) of \(G\) is said to be quasinormal in \(G\) if \(N K = K N\) for all subgroups \(K\) of \(G\); a Hall subgroup is a subgroup whose order is coprime to its index; and \(Z_{\mathfrak F}(L)\) is the \({\mathfrak F}\)-hypercenter of the group \(L\), where \(\mathfrak F\) is a formation. The main results of the paper under review are the following. Let \(G\) be the smallest prime divisor of the order of \(G\), and \(G_{p}\) be a Sylow \(p\)-subgroup of \(G\). If all of the maximal subgroups of \(G_{p}\) are \({\mathfrak U}_{h q}\)-supplemented in \(G\), then \(G\) is \(p\)-nilpotent. (Lemma~3.1) Here, \(\mathfrak U\) is the formation of supersolvable groups. Suppose \(G\) has a normal \(p\)-subgroup \(P\) such that \(G/P\) is supersolvable. If the maximal subgroups of \(P\) are \({\mathfrak U}_{hq}\)-supplemented in \(G\), then \(G\) itself is supersolvable (Theorem~3.3).
scientific article; zbMATH DE number 6909491