Abelian group

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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.

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.

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.

1. 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.
2. The identity element is typically denoted as  and termed zero
3. The inverse of an element is termed its negative or additive inverse. The inverse of  is denoted 
4.  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
`group`
either 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