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abelianization (Definition)

The abelianization of a group $ G$ is $ G/[G,G]$, the quotient 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, 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.



"abelianization" is owned by yark.
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See Also: derived subgroup

Other names:  abelianisation
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Cross-references: canonical projection, abelian group, homomorphic image, normal subgroup, abelian, derived subgroup, group
There are 10 references to this entry.

This is version 4 of abelianization, born on 2004-12-11, modified 2004-12-12.
Object id is 6561, canonical name is Abelianization.
Accessed 3591 times total.

Classification:
AMS MSC: 20F14 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Derived series, central series, and generalizations)
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