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

Let p bean 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 (sic)i (B) the subgroup of B generated by the p(i)-th powers of elements of B, that is (sic)i (B) = (sic)bpi | b is an element of B(sic). In this article we show that, for all values of i, the set A (sic)i (B) is a subgroup of G. We also present some applications of this result.