Powerful p-group

Definition
A powerful p-group is a p-group satisfying the condition below. The term is typically used only for finite p-groups, but the definition makes sense in the infinite context as well.

For the prime $$p = 2$$
A 2-group $$P$$ (i.e., a finite group of order a power of 2) is termed powerful if $$[P,P] \le \mho^2(P)$$ where $$[P,P]$$ is the defining ingredient::derived subgroup and $$\mho^2(P)$$ is the second agemo subgroup, i.e., the subgroup generated by the $$4^{th}$$ powers of elements.

For odd primes
Suppose $$P$$ is a $$p$$-group, $$p$$ an odd prime. $$P$$ is termed powerful if $$[P,P] \le \mho^1(P)$$ where $$[P,P]$$ is the defining ingredient::derived subgroup and $$\mho^1(P)$$ is the first agemo subgroup, i.e., the subgroup generated by all $$p^{th}$$ powers.