Abelian group
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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 is termed Abelian if for any elements and in , (here denotes the product of and in ).
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.
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. For full proof, refer: Abelianness is subgroup-closed
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. For full proof, refer: Abelianness is quotient-closed
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. For full proof, refer: Abelianness is direct product-closed
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
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
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 17 (definition as Point (2) in general definition of a group)
- Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 2 (definition introduced in paragraph)
- Topics in Algebra by I. N. Herstein, ^{More info}, Page 28 (formal definition)
- A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 2 (formal definition)
- Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, ^{More info}, Page 1 (definition introduced in paragraph)