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 termed abelian if it satisfies the following equivalent conditions:
- Its center is the whole group.
- Its derived subgroup is trivial.
- (Choose a generating set for ). For any elements , .
- The diagonal subgroup is a normal subgroup inside .
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 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
Weaker properties
Incomparable properties
- Supersolvable group is a group that has a normal series where all the successive quotient groups are cyclic groups. An abelian group is supersolvable if and only if it is finitely generated.
- Polycyclic group is a group that has a subnormal series where all the successive quotent groups are cyclic groups. An abelian group is polycyclic if and only if it is finitely generated.
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 978-0471433347^{More 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 0387945261^{More info} | 2 | 1.1 (Rudiments of Group Theory/Review) | definition introduced in paragraph | |
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632^{More info} | 42 | definition introduced in paragraph (immediately after definition of group) | ||
Topics in Algebra by I. N. Herstein^{More info} | 28 | Formal definition | ||
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info} | 2 | 1.1 (Binary Operations, Semigroups, and Groups) | formal definition | |
Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, Page 1, 1.1 (Elementary group theory), (definition introduced in paragraph)^{More info} |