abelian group

Let (G,*) be a group. If for any a,bG we have a*b=b*a, we say that the group is abelianMathworldPlanetmath (or commutativePlanetmathPlanetmathPlanetmath). Abelian groups are named after Niels Henrik Abel, but the word abelian is commonly written in lowercase.

Abelian groups are essentially the same thing as unitary -modules (http://planetmath.org/Module). In fact, it is often more natural to treat abelian groups as modules rather than as groups, and for this reason they are commonly written using additive notation.

Some of the basic properties of abelian groups are as follows:

Theorem 1.

Any subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) of an abelian group is normal.


Let H be a subgroup of the abelian group G. Since ah=ha for any aG and any hH we get aH=Ha. That is, H is normal in G. ∎

Theorem 2.

Quotient groupsMathworldPlanetmath of abelian groups are also abelian.


Let H be a subgroup of G. Since G is abelian, H is normal and we can get the quotient group G/H whose elements are the equivalence classesMathworldPlanetmathPlanetmath for ab if ab-1H. The operationMathworldPlanetmath on the quotient group is given by aHbH=(ab)H. But bHaH=(ba)H=(ab)H, therefore the quotient group is also commutative. ∎

Here is another theorem concerning abelian groups:

Theorem 3.

If φ:GG defined by φ(x)=x2 is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/GroupHomomorphism), then G is abelian.


If such a function were a homomorphism, we would have


that is, xyxy=xxyy. Left-multiplying by x-1 and right-multiplying by y-1 we are led to yx=xy and thus the group is abelian. ∎

Title abelian group
Canonical name AbelianGroup
Date of creation 2013-03-22 14:01:55
Last modified on 2013-03-22 14:01:55
Owner yark (2760)
Last modified by yark (2760)
Numerical id 25
Author yark (2760)
Entry type Definition
Classification msc 20K99
Synonym commutative group
Related topic Group
Related topic Klein4Group
Related topic CommutativeSemigroup
Related topic GeneralizedCyclicGroup
Related topic AbelianGroupsOfOrder120
Related topic FundamentalTheoremOfFinitelyGeneratedAbelianGroups
Related topic NonabelianGroup
Related topic Commutative
Related topic MetabelianGroup
Defines abelian
Defines commutative