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 component-wise 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:

 In:= Out= Permutation generators of an Abelian group:

 In:= Out= Elements of a permutation representation of an Abelian group:

 In:= Out= Neat Examples(1)

Introduced in 2010
(8.0)