Group Homology
In terms of group homology, a perfect group is precisely one whose first homology group vanishes: H1(G, Z) = 0, as the first homology group of a group is exactly the abelianization of the group, and perfect means trivial abelianization. An advantage of this definition is that it admits strengthening:
- A superperfect group is one whose first two homology groups vanish: H1(G, Z) = H2(G, Z) = 0.
- An acyclic group is one all of whose (reduced) homology groups vanish (This is equivalent to all homology groups other than H0 vanishing.)
Read more about this topic: Perfect Group
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