Omega subgroups of powerful <Emphasis Type="Italic">p</Emphasis>-groups

Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all i ≥ 0: (i) exp Ω_i(G) ≤ p ⁱ for odd p, and expΩ_i(G) ≤ 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. Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University of the Basque Country, grant UPV05/99.