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.