On the factorised subgroups of products of cyclic and non-cyclic finite p-groups

Let p be an odd prime and let G=AB be a finite p-group that is the product of a cyclic subgroup A and a non-cyclic subgroup B. Suppose in addition that the nilpotency class of B is less than p 2. We denote by ℧ i (B) the subgroup of B generated by the p i -th powers of elements of B, that is ℧ i (B)=〈b p i ∣b∈B〉. In this article we show that, for all values of i, the set A℧ i (B) is a subgroup of G. We also present some applications of this result.