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

Title	abelianization
Canonical name	Abelianization
Date of creation	2013-03-22 14:52:57
Last modified on	2013-03-22 14:52:57
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	7
Author	yark (2760)
Entry type	Definition
Classification	msc 20F14
Synonym	abelianisation
Related topic	DerivedSubgroup