Potent p-group

Definition
The term is typically used for a finite p-group, but we present the definition here for an arbitrary p-group.

Let $$p$$ be a prime number and $$G$$ be a p-group. We say that $$G$$ is potent if it satisfies the following.

Case $$p = 2$$
If $$p = 2$$, we require that $$[G,G] \le \mho^2(G)$$. In other words, the derived subgroup $$[G,G]$$ of $$G$$ is contained in the second agemo subgroup $$\mho^2(G)$$, i.e., the subgroup generated by fourth powers.

Case odd $$p$$
For odd $$p$$, we require that $$\gamma_{p-1}(G) \le \mho^1(G)$$. Here, $$\gamma_{p-1}(G)$$ is the $$(p-1)^{th}$$ member of the lower central series of $$G$$ and $$\mho^1(G)$$ is the first agemo subgroup: the subgroup generated by all the $$p^{th}$$ powers.