Group of prime power order whose derived subgroup is in the first agemo subgroup of its center

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Definition

A group of prime power order whose derived subgroup is in the first agemo subgroup of its center is a group of prime power order P, i.e., a finite p-group for some prime number p, such that the derived subgroup is contained in the first agemo subgroup of the center. In symbols:

[P,P] \le \mho^1(Z(P))

In other words, every commutator is the p^{th} power of some central element of the group.