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.

Subgroup properties satisfied

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

Related notions