# AbelianGroup

AbelianGroup[{n1,n2,}]

represents the direct product of the cyclic groups of degrees n1,n2,.

# Details

• The degrees ni of AbelianGroup[{n1,n2,}] must be non-negative integers.
• AbelianGroup[{n1,n2,}] is represented by default as a permutation group on the points {1,,n1+n2+}.

# Background & Context

• AbelianGroup[{n1,n2,,nk}] represents the commutative group defined as the direct product of cyclic groups having non-negative integer degrees n1,n2,,nk. Here, the direct product of groups , , is the analog of the Cartesian product of sets in which the underlying sets are ordered tuples with , , and the group operation is taken componentwise so that .
• In general, the term "Abelian group" is used to refer to a group that is commutative, i.e. a group for which the group operation satisfies the identity for all elements . The fundamental theorem of finite Abelian groups states that every finite Abelian group can be expressed as a direct product of cyclic groups. As a result, the function AbelianGroup can be used to represent any finite Abelian group.
• The default representation of AbelianGroup[{n1,n2,,nk}] is as a permutation group on the elements . When , AbelianGroup[{n}] is equivalent to CyclicGroup[n] (with both AbelianGroup[{0}] and AbelianGroup[{1}] equivalent to the trivial group with exactly one element).
• The usual group theoretic functions may be applied to AbelianGroup[{n1,n2,,nk}], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the Abelian group AbelianGroup[{n1,n2,,nk}] are available via FiniteGroupData[{"AbelianGroup",{n1,n2,,nk}},"prop"].
• AbelianGroup is related to a number of other symbols. Mathematically, AbelianGroup[{n1,n2,,nk}] is equivalent to the direct product of the groups CyclicGroup[n1],CyclicGroup[n2],,CyclicGroup[nk]. Other infinite families of finite groups built into the Wolfram Language that are parametrized by integers include AlternatingGroup, CyclicGroup, DihedralGroup and SymmetricGroup.

# Examples

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## Basic Examples(3)

Number of elements of an Abelian group:

Permutation generators of an Abelian group:

Elements of a permutation representation of an Abelian group:

## Scope(1)

Direct product of any number of cyclic groups of any degrees:

## Properties & Relations(2)

For nonzero degrees, the order of AbelianGroup[{n1,n2,}] is the product of the ni:

A group AbelianGroup[{p,p,}] with p prime is called an elementary Abelian group. In such groups all nontrivial elements have order p:

## Neat Examples(1)

The Cayley graph of the direct product of two cyclic groups resembles a torus:

Wolfram Research (2010), AbelianGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/AbelianGroup.html.

#### Text

Wolfram Research (2010), AbelianGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/AbelianGroup.html.

#### CMS

Wolfram Language. 2010. "AbelianGroup." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AbelianGroup.html.

#### APA

Wolfram Language. (2010). AbelianGroup. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AbelianGroup.html

#### BibTeX

@misc{reference.wolfram_2022_abeliangroup, author="Wolfram Research", title="{AbelianGroup}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/AbelianGroup.html}", note=[Accessed: 01-April-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_abeliangroup, organization={Wolfram Research}, title={AbelianGroup}, year={2010}, url={https://reference.wolfram.com/language/ref/AbelianGroup.html}, note=[Accessed: 01-April-2023 ]}