AlternatingGroup

AlternatingGroup[n]

represents the alternating group of degree n.

Details

The degree n of AlternatingGroup[n] must be a non-negative integer. Degrees 0, 1, and 2 correspond to the trivial or identity group.
AlternatingGroup[n] is represented by default as a permutation group on the points {1,…,n}.

Background & Context

AlternatingGroup[n] represents the alternating group (sometimes denoted ) on n symbols for a given non-negative integer n. For , the default representation of AlternatingGroup[n] is as a permutation group on the symbols . The special cases AlternatingGroup[0], AlternatingGroup[1] and AlternatingGroup[2] are equivalent to the trivial group with exactly one element.
Mathematically, the alternating group (for ) consists of the even permutations of the symbols (i.e. those permutations of having permutation signature ), together with the group operation of composition. Alternating groups are therefore permutation groups of order , with AlternatingGroup[n] isomorphic to PermutationGroup[perms], where perms=Select[Permutations[Range[n]],Signature[#]1&].
Alternating groups are of fundamental importance in abstract algebra, geometric group theory, representation theory, combinatorics and mathematical physics. A number of important mathematical results hold for alternating groups. For example, by noting that every element of can be written as a composition of 3-cycles (permutations whose length equals 3), it follows that is simple (i.e. has no nontrivial normal subgroups) for , a fact that played a pivotal role in the classification of finite simple groups. In addition, there are a number of exceptional isomorphisms between the alternating groups (for n small) and certain small groups of Lie type. Finally, the group homology of alternating groups is known to be stable, in the sense that it becomes constant as n grows large.
The usual group theoretic functions may be applied to AlternatingGroup[n], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the alternating group are available via FiniteGroupData[{"AlternatingGroup",n},"prop"].
AlternatingGroup is related to a number of other symbols. Both alternating groups and dihedral groups are important subgroups of symmetric groups. Other infinite families of finite groups built into the Wolfram Language that are parametrized by integers include AbelianGroup, CyclicGroup, DihedralGroup and SymmetricGroup.