Superperfect Group - Definition


The first homology group of a group is the abelianization of the group itself, since the homology of a group G is the homology of any Eilenberg-MacLane space of type K(G, 1); the fundamental group of a K(G, 1) is G, and the first homology of K(G, 1) is then abelianization of its fundamental group. Thus, if a group is superperfect, then it is perfect.

A finite perfect group is superperfect if and only if it is its own universal central extension (UCE), as the second homology group of a perfect group parametrizes central extensions.

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