This article defines a group property that is pivotal (i.e., important) among existing group properties
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This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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.
An Abelian group is a group where any two elements commute.
Definition with symbols
A group is termed Abelian if for any elements and in , (here denotes the product of and in ).
- A group is Abelian if its center is the whole group.
- A group is Abelian if its commutator subgroup is trivial.
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.
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.
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.
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
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
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
The testing problem
Further information: Abelianness testing problem
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:
groupeither defines the group or gives the name to a group previously defined.
Study of this notion
Mathematical subject classification
Under the Mathematical subject classification, the study of this notion comes under the class: 20K