Characterizations of hypercyclically embedded subgroups of finite groups. (Q2630375)

Summary: A normal subgroup \(H\) of a finite group \(G\) is said to be \textit{hypercyclically embedded in} \(G\) if every chief factor of \(G\) below \(H\) is cyclic. Our main goal here is to give new characterizations of hypercyclically embedded subgroups. In particular, we prove that a normal subgroup \(E\) of a finite group \(G\) is hypercyclically embedded in \(G\) if and only if for every different primes \(p\) and \(q\) and every \(p\)-element \(a\in(G'\cap F^*(E))E'\), \(p'\)-element \(b\in G\) and \(q\)-element \(c\in G'\) we have \([a,b^{p-1}]=1=[a^{q-1},c]\). Some known results are generalized.