Editing Perfect group (section)

==Group homology==
In terms of [[group homology]], a perfect group is precisely one whose first homology group vanishes: ''H''<sub>1</sub>(''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: <math>H_1(G,\Z)=H_2(G,\Z)=0</math>.
* An [[acyclic group]] is one ''all'' of whose (reduced) homology groups vanish <math>\tilde H_i(G;\Z) = 0.</math> (This is equivalent to all homology groups other than <math>H_0</math> vanishing.)