Agemo subgroups of a p-group

Definition
Let $$p$$ be a prime number and $$P$$ be a $$p$$-group. For any nonnegative integer $$j$$, the $$j^{th}$$ agemo subgroup of $$P$$ is defined as:

$$\mho^j(P) = \langle x^{p^j} \mid x \in P \rangle $$

In other words, it is the subgroup generated by all the $$(p^j)^{th}$$ powers.

If the exponent of $$P$$ is $$p^r$$, then $$\mho^r(P)$$ (and any higher agemo subgroup) is trivial, and all previous $$\mho^j(P)$$ are nontrivial.

The subgroups form a descending chain.

We can also consider agemo subgroups of a pro-p-group.

Subgroup properties satisfied
All the agemo subgroups are fully invariant subgroups, in fact, they're all verbal subgroups.

Related notions

 * Iterated agemo subgroups of a p-group