February 22, 2008
We prove that the exponent Ω m ( G ) is at most p m if G is a powerful p -group with p odd. Calling on a recent result of Héthelyi and Lévai, we prove that | G p m | = | G :Ω 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.