On orders and vanishing of integral cohomology groups

Let G be a finite group with a non-trivial normal subgroup N such that G/N is cyclic. In this paper we obtain a recurrence relation involving the orders of the integral cohomology groups Hn(G,Z), n⩾1. We then use this result to show that no two consecutive integral cohomology groups of G can vanish. This in turn is used to show that if G is a finite group and k is a field of characteristic p such that H1(G,k)≠0, then H2n(G,k)≠0 for all n⩾0. Some results relating the restriction and corestriction maps in integral cohomology are also obtained.