Abelian group
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.
The word abelian is usually begun with a small a.
wikinote: Some older content on the wiki uses capital A for Abelian. We're trying to update this content.
Definition
An abelian group is a group where any two elements commute. In symbols, a group is termed abelian if for any elements and in , (here denotes the product of and in ). Note that are allowed to be equal, though equal elements commute anyway, so we can restrict attention if we wish to unequal elements.
Full definition
An abelian group is a set equipped with a (infix) binary operation (called the addition or group operation), an identity element and a (prefix) unary operation , called the inverse map or negation map, satisfying the following:
- For any , . This property is termed associativity.
- For any , . thus plays the role of an additive identity element or neutral element.
- For any , . Thus, is an inverse element to with respect to .
- For any , . This property is termed commutativity.
Equivalent formulations
- A group is abelian if its center is the whole group.
- A group is abelian if its derived subgroup is trivial.
Notation
When is an abelian group, we typically use additive notation and terminology. Thus, the group multiplication is termed addition and the product of two elements is termed the sum.
- The infix operator is used for the group multiplication, so the sum of two elements and is denoted by . The group multiplication is termed addition and the product of two elements is termed the sum.
- The identity element is typically denoted as and termed zero
- The inverse of an element is termed its negative or additive inverse. The inverse of is denoted
- done times is denoted , (where ) while done times is denoted .
This convention is typically followed in a situation where we are dealing with the abelian group in isolation, rather than as a subgroup of a possibly non-abelian group. If we are working with subgroups in a non-abelian group, we typically use multiplicative notation even if the subgroup happens to be abelian.
Examples
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Some infinite examples
The additive group of integers , the additive group of rational numbers , the additive group of real numbers , the multiplicative group of nonzero rationals , and the multiplicative group of nonzero real numbers are some examples of Abelian groups.
(More generally, for any field, the additive group, and the multiplicative group of nonzero elements, are Abelian groups).
Finite examples
Cyclic groups are good examples of abelian groups, where the cyclic group of order is the group of integers modulo .
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.
Non-examples
Not every group is abelian. The smallest non-abelian group is the symmetric group on three letters: the group of all permutations on three letters, under composition. Its being non-abelian hinges on the fact that the order in which permutations are performed, matters.
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 an abelian-quotient subgroup (i.e., normal with abelian quotient group) if and only if the subgroup contains the commutator subgroup.
Formalisms
In terms of the diagonal-in-square operator
This property is obtained by applying the diagonal-in-square operator to the property: normal subgroup
View other properties obtained by applying the diagonal-in-square operator
A group is an abelian group if and only if, in the external direct product , the diagonal subgroup is a normal subgroup.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | pProof of strictness (reverse implication failure) | Intermediate notions | Comparison |
---|---|---|---|---|---|
cyclic group | generated by one element | cyclic implies abelian | abelian not implies cyclic (see also list of examples) | For intermediate notions between abelian group and cyclic group, click here. | |
homocyclic group | direct product of isomorphic cyclic groups | (see also list of examples) | |||
finite abelian group | abelian and a finite group | (see also list of examples) | |||
finitely generated abelian group | abelian and a finitely generated group | (see also list of examples) |
Weaker properties
Property | Meaning | Proof of implication | pProof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
nilpotent group | lower central series reaches identity, upper central series reaches whole group | abelian implies nilpotent | nilpotent not implies abelian (see also list of examples) | For intermediate notions between nilpotent group and abelian group, click here. |
solvable group | derived series reaches identity, has normal series with abelian factor groups | abelian implies solvable | solvable not implies abelian (see also list of examples) | For intermediate notions between solvable group and abelian group, click here. |
metabelian group | has abelian normal subgroup with abelian quotient group | (see also list of examples) | ||
virtually abelian group | has abelian subgroup of finite index |
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).
The GAP command for this group property is:IsAbelian
The class of all groups with this property can be referred to with the built-in command: AbelianGroups
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)
- Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 42 (defined immediately after the definition of group, as a group where the composition is commutative)
- 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)