Tour:Abelian group

The notion of Abelian group is very important. Abelian groups are those groups where the binary operation is commutative. Read, and thoroughly understand, the definition of Abelian group given below, and then proceed.

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Definition

Symbol-free definition

An Abelian group is a group where any two elements commute.

Definition with symbols

A group $G$ is termed Abelian if for any elements $x$ and $y$ in $G$, $xy=yx$.

Equivalent formulations

• A group is Abelian if its center is the whole group.
• A group is Abelian if its commutator subgroup is trivial.

Examples

Cyclic groups are good examples of Abelian groups. Further, any direct product of cyclic groups is also an Abelian group. Further, every finitely generated Abelian group is obtained this way. This is the famous structure theorem for finitely generated Abelian groups.

The structure theorem can be used to generate a complete listing of finite Abelian groups, as described here: classification of finite Abelian groups.