Finite groups in which primary subgroups have cyclic cofactors. (Q538034)

The authors prove the following theorem: Let \(G\) be a finite group, \(q\) be the largest prime divisor of \(|G|\) and \(\pi=\pi(G)\setminus\{q\}\). Suppose that the factor group \(X/\text{core}_GX\) is cyclic for every \(p\)-subgroup \(X\) of \(G\) and every \(p\in\pi\). Then: (1) \(G\) is solvable and its Hall \(\{2,3\}'\)-subgroup is normal in \(G\) and is a dispersive group of Ore; (2) All Hall \(\{2,3\}\)-subgroups of \(G\) are meta-nilpotent; (3) Every Hall \(p'\)-subgroup of \(G\) is a dispersive group of Ore, for every \(p\in\{2,3\}\); (4) \(l_r(G)\leq 1\), for all \(r\in\pi(G)\).