Agemo subgroups of a p-group

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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.

Name origin

The word "agemo" is "omega" backwards, relating to how the symbol $\mho$ used for these groups is $\Omega$ upside-down. They are related to the Omega subgroups of a 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