abelianization
The abelianization of a group is , the quotient (http://planetmath.org/QuotientGroup) of by its derived subgroup.
The abelianization of is the largest abelian quotient of , in the sense that if is a normal subgroup of then is abelian if and only if . In particular, every abelian quotient of is a homomorphic image of .
If is an abelian group and is a homomorphism (http://planetmath.org/GroupHomomorphism), then there is a unique homomorphism such that , where 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 |