\(\mathscr{H}C\)-subgroups and the \(p\mathfrak{F}\)-hypercenter of finite groups (Q6635908)

A subgroup \(H\) of a group \(G\) is called an \(HC\)-subgroup of \(G\) if there exists a normal subgroup \(T\) of \(G\) such that \(G = HT\) and \(H^{g} \cap N_{T }(H) \leq H\) for all \(g \in G\). In this paper, the authors obtained a interesting result for a normal subgroup to be contained in the \(pF\)-hypercenter of a finite group by assuming that some of its subgroups are \(H C\)-subgroups. Those results generalize and uniform many known results.