Finite solvable groups in which the Sylow p-subgroups are either bicyclic or of order p 3

Abstract

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p 3 for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A4-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p 2 + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.