Superperfect group
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In mathematics, in the realm of group theory, a group is said to be superperfect when its first two homology groups are trivial.
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.
For example, if G is the fundamental group of a homology sphere, then G is superperfect. The smallest finite, non-trivial superperfect group is the binary icosahedral group (the fundamental group of the Poincaré homology sphere).
Every acyclic group is superperfect, but the converse is not true: the binary icosahedral group is superperfect, but not acyclic.
