Powerful p-group

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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.