Powerful pro-p-group

Definition
A powerful pro-p-group is a pro-p-group satisfying the condition below.

For the prime $$p = 2$$
For the prime $$p = 2$$, a pro-$$p$$-group 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 fourth powers of elements.

For odd primes
Suppose $$P$$ is a pro-$$p$$-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.