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.

单词	Abelianization
释义	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.
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