Superperfect group

Superperfect group

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.

References

* A. Jon Berrick and Jonathan A. Hillman, "Perfect and acyclic subgroups of finitely presentable groups", Journal of the London Mathematical Society (2) 68 (2003), no. 3, 683--698. MathSciNet|id=2009444


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