Perfect group
A group G is perfect if it equals its own derived subgroup:
- G = [G, G] (equivalently G⁽¹⁾ = G).
- Equivalently, its abelianization is trivial: Gᵃᵇ = G/[G, G] = {e}.
- Equivalently: G has no nontrivial abelian quotient.
Key facts
- Every nonabelian simple group is perfect.
- Every quasisimple group is perfect (perfect is part of the definition of quasisimple).
- The only abelian perfect group is the trivial group {e}.
- A nontrivial perfect group is never solvable.
- The derived series G ▷ G⁽¹⁾ ▷ G⁽²⁾ ▷ ⋯ stabilizes at the perfect core: the largest perfect subgroup of G.
- Perfect is exactly H₁(G; ℤ) = 0. When H₂(G; ℤ) = M(G) vanishes too, G is superperfect.
Examples
A₅, order 60 — the smallest nontrivial perfect group; it is simple.SL(2,𝔽5), order 120 — perfect and quasisimple, but not simple (center ≅ C₂). It is also superperfect: M(SL(2,𝔽5)) = {e}.
Non-examples
- No nontrivial solvable group is perfect: every solvable group has nontrivial abelianization.
- In particular every nontrivial abelian, nilpotent, or supersolvable group is not perfect.