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
Famous quotes containing the word group:
“Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
—Simone Weil (1910–1943)