Abelian group

This article defines a group property that is pivotal (i.e., important) among existing group properties
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VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions<random> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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History

Origin of the term

The term Abelian group comes from Niels Henrick Abel, a mathematician who worked with groups even before the formal theory was laid down, in order to prove unsolvability of the quintic.

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.

Facts

Occurrence as subgroups

Every cyclic group is Abelian. Since each group is generated by its cyclic subgroups, every group is generated by a family of Abelian subgroups. A trickier question is: do there exist Abelian normal subgroups? A good candidate for an Abelian normal subgroup is the center, which is the collection of elements of the group that commute with every element of the group.

Occurrence as quotients

The maximal Abelian quotient of any group is termed its Abelianization, and this is the quotient by the commutator subgroup. A subgroup is normal with Abelian quotient group if and only if the subgroup contains the commutator subgroup.

Metaproperties

Varietal group property

This group property is a varietal group property, in the sense that the collection of groups satisfying this property forms a variety of algebras. In other words, the collection of groups satisfying this property is closed under taking subgroups, taking quotients and taking arbitrary direct products.

Abelian groups form a variety of algebras. The defining equations for this variety are the equations for a group along with the commutativity equation.

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Any subgroup of an Abelian group is Abelian -- viz the property of being Abelian is subgroup-closed. This follows as a direct consequence of Abelianness being varietal.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of an Abelian group is Abelian -- viz the property of being Abelian is quotient-closed. This again follows as a direct consequence of Abelianness being varietal.

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of Abelian groups is Abelian -- viz the property of being Abelian is direct product-closed. This again follows as a direct consequence of Abelianness being varietal.

Testing

The testing problem

Further information: Abelianness testing problem

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
View GAP-testable group properties

To test whether a group is Abelian, the GAP syntax is:

IsAbelian (group)

where

group

either defines the group or gives the name to a group previously defined.

External links

Definition links

• Wikipedia page
• Planetmath page
• Mathworld page
• Springer Online Reference page

Perspective links

• Chapman page (from the algebraic structures viewpoint)