Finite Groups in which Primary Subgroups have Cyclic Cofactors

In this paper, we prove the following theorem: Let G be a group, q be the largest prime divisor of |G| and (pi) = (pi)(G) {q}. Suppose that the factor group X/coreGX is cyclic for every p-subgroup X of G and every p (in) (pi). Then: (1) G is soluble and its Hall {2,3}´-subgroup is normal in G and is a dispersive group by Ore; (2) All Hall {2,3}-subgroups of G are metanilpotent; (3) Every Hall p´-subgroup of G is a dispersive group by Ore, for every p (in) {2,3}; (4) lr(G) ≤ 1, for all r (in) (pi)(G).