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Commutator SubgroupEdit

In the study of algebra, the commutator subgroup is a central construct that encodes how far a given group is from being commutative. It is the subgroup generated by all commutators [g,h] = g^{-1}h^{-1}gh, where g and h range over the group. This subgroup is always normal, and the quotient by it carries a universal abelian flavor: G/[G,G] is the largest abelian quotient of G, often called the Abelianization of the group. The idea is simple but powerful: by peeling away the noncommutative layers via the commutator subgroup, one arrives at a clean abelian shadow of the original structure.

Beyond the basic definition, the commutator subgroup appears in a broad web of ideas across mathematics. It provides a bridge between non-abelian symmetry and abelian invariants, and it features prominently in the study of how a group acts on other objects, from geometric spaces to algebraic structures. The concept also underpins many constructions in topology, number theory, and representation theory, where understanding the abelian quotient can illuminate the ways a noncommutative object can be simplified or classified.

Definition

Let G be a group. For elements g,h in G, the commutator is defined by [g,h] = g^{-1}h^{-1}gh. The commutator subgroup, denoted [G,G] or G', is the subgroup generated by all such commutators: - [G,G] = ⟨ [g,h] : g,h ∈ G ⟩.

This subgroup is normal in G, and the quotient G/[G,G] is abelian. The abelianization functorially captures all abelian images of G: any homomorphism from G to an Abelian group factors uniquely through G/[G,G]. In symbols, if φ: G → A with A abelian, then there exists a unique φ̄: G/[G,G] → A with φ = φ̄ ∘ π, where π: G → G/[G,G] is the natural projection.

Basic properties

  • Normality: [G,G] is a normal subgroup of G. This ensures the quotient G/[G,G] makes sense as a well-behaved quotient group.
  • Characterization of abelian quotients: G/[G,G] is the largest abelian quotient of G; equivalently, G' ≤ N if and only if G/N is abelian.
  • Trivial case: [G,G] = {e} if and only if G is abelian.
  • Functorial behavior: The construction G ↦ G/[G,G] is functorial in the sense that homomorphisms out of G induce homomorphisms out of the abelianization.
  • Relation to other subgroups: Although [G,G] is generated by commutators, it is not in general contained in the center Z(G); the two notions reflect different aspects of noncommutativity.

Examples

  • Symmetric groups: For the symmetric group Symmetric group with n ≥ 3, the derived subgroup is the alternating group Alternating group. Consequently, S_n/[S_n,S_n] ≅ C2, reflecting the sign homomorphism.
  • Heisenberg group: The Heisenberg group H(Z) of integer upper triangular 3×3 matrices with ones on the diagonal has a commutator subgroup equal to its center, consisting of matrices with a nonzero upper-right entry and zeros elsewhere. This makes the abelianization isomorphic to Z ⊕ Z.
  • Finite abelian groups: If G is already abelian, then [G,G] = {e}, so the abelianization is G itself.
  • Free groups: If F is a free group on a set S, then its abelianization is the free abelian group on S, often denoted ⊕_S Z.

Derived series, solvability, and related notions

The commutator subgroup is the first step in the derived series of a group: - G^{(0)} = G - G^{(n+1)} = [G^{(n)}, G^{(n)}]

A group is called solvable if its derived series terminates in the trivial subgroup after finitely many steps. The concept of the commutator subgroup is thus central to the study of how noncommutative a group is and how readily its structure can be broken into successive abelian layers. A related construction is the lower central series, which measures a stronger form of noncommutativity and leads to the notion of nilpotent groups.

Computation and presentations

In practical terms, computing the abelianization of a group given by a presentation ⟨S | R⟩ involves imposing commutativity on the generators and enforcing the original relations. Concretely, the abelianization is presented by taking S as a generating set of a free abelian group and adding the relations R with all generators commuting: - G^{ab} ≅ ⟨S | R, [s_i, s_j] for all i ≠ j⟩, where [s_i, s_j] denotes the commutator of generators s_i and s_j.

This approach lets one compute abelian invariants of many groups, including finitely presented examples, and it underpins algorithms implemented in computational algebra systems for determining abelianizations.

Applications and connections

  • Topology: There is a foundational link between the abelianization of the Fundamental group π1(X) and the first homology group H1(X). In particular, H1(X) ≅ π1(X)^{ab}, tying noncommutative topology to abelian invariants.
  • Galois theory: The abelianization of a Galois group corresponds to the maximal abelian extension in certain contexts, reflecting a deep connection between noncommutative symmetry and abelian structures.
  • Representation theory: Understanding the abelianization of a group can simplify the study of one-dimensional representations, since characters factor through the abelianization.
  • Classical groups: The behavior of [G,G] in classical groups (e.g., Special linear group versus General linear group quotients) sheds light on how noncommutativity manifests in matrix groups.

See also