abelian group

Let $(G,*)$ be a group. If for any $a,b\in G$ we have $a*b=b*a$, we say that the group is abelian (or commutative). 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 $\mathbb{Z}$-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 subgroup (http://planetmath.org/Subgroup) of an abelian group is normal.

Proof.

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

Theorem 2.

Quotient groups of abelian groups are also abelian.

Proof.

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 classes for $a\sim b$ if $a{b}^{-1}\in H$. The operation on the quotient group is given by $aH\cdot bH=(ab)H$. But $bH\cdot aH=(ba)H=(ab)H$, therefore the quotient group is also commutative. ∎

Here is another theorem concerning abelian groups:

Theorem 3.

If $\phi \mathrm{:}G\mathrm{\to }G$ defined by $\phi \mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}{x}^{\mathrm{2}}$ is a homomorphism (http://planetmath.org/GroupHomomorphism), then $G$ is abelian.

Proof.

If such a function were a homomorphism, we would have

$${(xy)}^2=\phi (xy)=\phi (x)\phi (y)=x^2{y}^2$$

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