# Powerful p-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

A powerful p-group is a p-group satisfying the condition below. The term is typically used only for finite p-groups, but the definition makes sense in the infinite context as well.

### For the prime $p = 2$

A 2-group $P$ (i.e., a finite group of order a power of 2) is termed powerful if $[P,P] \le \mho^2(P)$ where $[P,P]$ is the derived subgroup and $\mho^2(P)$ is the second agemo subgroup, i.e., the subgroup generated by the $4^{th}$ powers of elements.

### For odd primes

Suppose $P$ is a $p$-group, $p$ an odd prime. $P$ is termed powerful if $[P,P] \le \mho^1(P)$ where $[P,P]$ is the derived subgroup and $\mho^1(P)$ is the first agemo subgroup, i.e., the subgroup generated by all $p^{th}$ powers.