In general, groups are not Abelian. However, there is always a group homomorphism h:G->G^' to an Abelian group, and this homomorphism is called Abelianization. The homomorphism is abstractly described by its kernel, the commutator subgroup [G,G], which is the unique smallest normal subgroup of G such that the quotient group G^'=G/[G,G] is Abelian. Roughly speaking, in any expression, every product becomes commutative after Abelianization. As a consequence, some previously unequal expressions may become equal, or even represent the identity element.

For example, in the eight-element quaternion group G={+/-1,+/-i,+/-j,+/-k}, the commutator subgroup is {+/-1}. The Abelianization of G is a copy of Z_2×Z_2, and for instance, i^'j^'=j^'i^' in the Abelianization.

See also

Abelian, Commutator Subgroup, Group, Homomorphism

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Abelianization." From MathWorld--A Wolfram Web Resource.

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