Commutator subgroup

The commutator subgroup measures how far a group is from being abelian. Its quotient, the abelianization, is the largest abelian quotient of G.

Definition

For a, b ∈ G, the commutator of a and b is

[a, b]  =  a⁻¹ b⁻¹ a b.

(Conventions differ — some authors use aba⁻¹b⁻¹. The choice does not change the generated subgroup.) Note [a, b] = e ⇔ ab = ba.

The commutator subgroup (or derived subgroup) is

[G, G]  =  G'  =  ⟨ [a, b]  :  a, b ∈ G ⟩.

The abelianization of G is the quotient Gab = G / [G, G].

Properties

1. Normal. G' ⊴ G, and in fact G' is characteristic.
2. Smallest abelian quotient. For any normal subgroup N ⊴ G, G/N is abelian iff N ⊇ G'. In particular G / G' is abelian.
3. G' = {e} ⇔ G abelian.
4. Universal property. Every homomorphism G → A to an abelian group factors uniquely through G → Gab → A.
5. Commutator identities. [a, b]⁻¹ = [b, a], [ab, c] = [a, c]b · [b, c], [a, bc] = [a, c] · [a, b]c.

Examples

• [Sₙ, Sₙ] = Aₙ — the commutator subgroup of the symmetric group is the alternating group.
• [D₄, D₄] = {e, r²} — the central subgroup of order 2.
• [Q₈, Q₈] = {±1} — the center of Q₈.
• [GL(n,F), GL(n,F)] = SL(n,F) for most fields (|F| > 3 suffices).
• Abelianization of Sₙ: Sₙab ≅ C₂ via the sign map.
• Abelianization of Dₙ: C₂ if n odd, C₂ × C₂ if n even.

Computation

• Polycyclic group. G' is generated by the pc commutators [aᵢ, aⱼ] for i < j, where a₁, …, aₙ are the pc generators. Polynomial in n.
• Abelianization. The abelianization of a pc group of order ∏ pᵢeᵢ is read off as an abelian group via the relation matrix; Smith normal form converts to invariant factors (see abelian).
• Permutation group. G' is the normal closure of {[xᵢ, xⱼ]} for generators xᵢ, xⱼ; computed by normal closure in a BSGS.

See also

• Center — the "most abelian" subgroup; dual concept.
• Abelian group — G / G' is always abelian.
• Derived series — iterate: G ⊇ G' ⊇ G'' ⊇ ….
• Solvable group — derived series reaches {e}.
• Wikipedia: Commutator subgroup