# abelianization

The abelianization of a group $G$ is $G/[G,G]$, the quotient (http://planetmath.org/QuotientGroup) of $G$ by its derived subgroup.

The abelianization of $G$ is the largest abelian quotient of $G$, in the sense that if $N$ is a normal subgroup of $G$ then $G/N$ is abelian if and only if $[G,G]\subseteq N$. In particular, every abelian quotient of $G$ is a homomorphic image of $G/[G,G]$.

If $A$ is an abelian group and $\phi\colon G\to A$ is a homomorphism (http://planetmath.org/GroupHomomorphism), then there is a unique homomorphism $\psi\colon G/[G,G]\to A$ such that $\psi\circ\pi=\phi$, where $\pi\colon G\to G/[G,G]$ is the canonical projection.

Title abelianization Abelianization 2013-03-22 14:52:57 2013-03-22 14:52:57 yark (2760) yark (2760) 7 yark (2760) Definition msc 20F14 abelianisation DerivedSubgroup