Abelian group

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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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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:

Equivalent formulations

A group is termed abelian if it satisfies the following equivalent conditions:


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 derived subgroup. A subgroup is an abelian-quotient subgroup (i.e., normal with abelian quotient group) if and only if the subgroup contains the derived subgroup.

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
varietal group property Yes The collection of abelian groups forms a subvariety of the variety of groups. In particular, it is closed under taking subgroups, quotients, and arbitrary direct products
subgroup-closed group property Yes abelianness is subgroup-closed If is an abelian group and is a subgroup of , then is abelian.
quotient-closed group property Yes abelianness is quotient-closed If is an abelian group and is a normal subgroup of , the quotient group is abelian.
direct product-closed group property Yes abelianness is direct product-closed Suppose , are abelian groups. Then, the external direct product is also abelian.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof 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) Epabelian group, Locally cyclic group, Residually cyclic group|FULL LIST, MORE INFO
homocyclic group direct product of isomorphic cyclic groups (see also list of examples) |
residually cyclic group every non-identity element is outside a normal subgroup with a cyclic quotient group |
locally cyclic group every finitely generated subgroup is cyclic (see also list of examples) Epabelian group|FULL LIST, MORE INFO
epabelian group abelian group whose exterior square is the trivial group |
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) Residually cyclic group|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof 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) Group in which class equals maximum subnormal depth, Group of nilpotency class three, Group of nilpotency class two, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, UL-equivalent group|FULL LIST, MORE INFO
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) Metabelian group, Metanilpotent group, Nilpotent group|FULL LIST, MORE INFO
metabelian group has abelian normal subgroup with abelian quotient group (see also list of examples) Group of nilpotency class two|FULL LIST, MORE INFO
virtually abelian group has abelian subgroup of finite index FZ-group|FULL LIST, MORE INFO
FZ-group center has finite index |
FC-group every conjugacy class is finite FZ-group, Group with finite derived subgroup|FULL LIST, MORE INFO

Incomparable properties

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.

Testing

The testing problem

Further information: Abelianness testing problem

The abelianness testing problem is the problem of testing whether a group (described using some group description rule, such as an encoding of a group or a multi-encoding of a group) is abelian.

Algorithms for the abelianness testing problem range from the brute-force black-box group algorithm for abelianness testing (that involves testing for every pair of elements whether they commute, and is quadratic in the order of the group) to the generating set-based black-box group algorithm for abelianness testing (that involves testing only on a generating set, and is quadratic in the size of the generating set).

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

References

Textbook references

Book Page number Chapter and section Contextual information View
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info 17 Formal definition (definition as point (2) in general definition of group)
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info 2 1.1 (Rudiments of Group Theory/Review) definition introduced in paragraph Google Books
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632More info 42 definition introduced in paragraph (immediately after definition of group)
Topics in Algebra by I. N. HersteinMore info 28 Formal definition
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info 2 1.1 (Binary Operations, Semigroups, and Groups) formal definition Google Books
Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754More info 1 1.1 (Elementary group theory) definition introduced in paragraph Google Books

External links

Definition links

Perspective links