# Difference between revisions of "Abelian group"

This article defines a group property that is pivotal (i.e., important) among existing group properties
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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 $G$, $xy = yx$.

## 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.