On the power structure of powerful p-groups

We prove that the exponent of Omega(m)(G) is at most p(m) if G is a powerful p-group with p odd. Calling on a recent result of Hethelyi and Levai, we prove that \G(Pm)\ = \G : Omega(m)(G)\ for all m. These results also hold for regular p-groups. We also bound the nilpotence class of a subgroup of a powerful group by e + 1, where p(e) is the exponent of the subgroup. This is just one more than what the bound would be if the subgroup were itself powerful.