abelianization Abelianization 2004-12-11 08:21:04 2004-12-12 14:16:34 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{homomorphism} \PMlinkescapeword{quotient} The \emph{abelianization} of a group $G$ is $G/[G,G]$, the \PMlinkname{quotient}{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 \PMlinkname{homomorphism}{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.