The derived subgroup \(G'\) collects all noncommutativity in a group. Quotienting by it gives the largest abelian quotient, so it has both a generator definition and a universal characterization. [@isaacs2008]
Definition
For subgroups \(H,K\), \([H,K]\) is generated by all \([h,k]\), and \(G'=[G,G]\). Alternative commutator conventions change formulas but not the generated subgroup.
Core result
Cosets commute in \(G/G'\) because \(xy\) and \(yx\) differ by a commutator. If \(G/N\) is abelian, every \([x,y]\) lies in \(N\), hence \(G'\leq N\). This also proves the universal factorization.
Structural properties
- \(G'\operatorname{char}G\) because automorphisms send commutators to commutators.
- \(G\) is abelian exactly when \(G'=1\), and perfect exactly when \(G'=G\).
- Homomorphic images satisfy \(\varphi(G')=\varphi(G)'\); for a subgroup one generally has only \(H'\leq G'\).
- If \(G/N\) is abelian, then \(G'\leq N\); central quotients make higher commutators control the remaining structure.
- Every transfer homomorphism kills \(G'\) because its target is abelian.
The derived subgroup packages all information visible to maps into abelian groups and is the common entry point for derived series, abelianization, and transfer.
Example and boundary
Knowledge network
The commutator core article supplies identities, the derived series iterates the construction, and abelianization states the universal property.
Proof and sources
Definitions, minimum-kernel property, and homomorphism behavior are in Section 4A and the appendix; the short proof is retained here. [@isaacs2008]
