Omega subgroups of powerful p-groups

Abstract

Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all $i\ge 0$: (i) $\exp Ω_-i(G)\le p^i$ for odd p, and $\exp Ω_-i(G)\le 2^{i+1}$ for p = 2; (ii) the index $|G:G^{p^i}|$ coincides with the number of elements of G of order at most $p^i$.