On elements of order p in powerful p-groups

Abstract

We investigate the set Ω {1}(P) of elements of order at most p in a powerful p-group P and prove that |Ω {1}(P)|=|P:P p| . As a corollary, we obtain a necessary and sufficient condition for Ω 1(P) to be of exponent p. We give an example to show that for p=2 there is a powerful 2-group such that Ω 1(P) is not of exponent 2.