AbelianGroup

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AbelianGroup[{n₁,n₂,…}]

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

Details

The degrees n_i 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,…,n_k}] represents the commutative group defined as the direct product of cyclic groups having non-negative integer degrees n₁,n₂,…,n_k. 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,…,n_k}] 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,…,n_k}], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the Abelian group AbelianGroup[{n₁,n₂,…,n_k}] are available via FiniteGroupData[{"AbelianGroup",{n₁,n₂,…,n_k}},"prop"].
AbelianGroup is related to a number of other symbols. Mathematically, AbelianGroup[{n₁,n₂,…,n_k}] is equivalent to the direct product of the groups CyclicGroup[n₁],CyclicGroup[n₂],…,CyclicGroup[n_k]. Other infinite families of finite groups built into the Wolfram Language that are parametrized by integers include AlternatingGroup, CyclicGroup, DihedralGroup and SymmetricGroup.