AlternatingGroup
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.
Examples
open all close allBasic Examples (3)
Number of elements of an alternating group:
AlternatingGroup[10]//GroupOrderPermutation generators of an alternating group:
AlternatingGroup[10]//GroupGeneratorsElements of a permutation representation of an alternating group:
AlternatingGroup[4]//GroupElementsScope (1)
Alternating groups of degree 0, 1, or 2 are the trivial group, only containing the identity:
AlternatingGroup[0]//GroupElementsAlternatingGroup[1]//GroupElementsAlternatingGroup[2]//GroupElementsIn all other cases the alternating group of degree n contains n!/2 elements:
AlternatingGroup[3]//GroupElementsAlternatingGroup[4]//GroupElementsApplications (2)
Test whether two random permutations generate the alternating group of degree 100:
SeedRandom[123];
g1 = RandomPermutation[100];
g2 = RandomPermutation[100];
PermutationGroup[{g1, g2}] == AlternatingGroup[100]Permute parts of an expression under the elements of an alternating group:
Permute[F[a, b, c, d, e], AlternatingGroup[4]]Text
Wolfram Research (2010), AlternatingGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/AlternatingGroup.html.
CMS
Wolfram Language. 2010. "AlternatingGroup." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AlternatingGroup.html.
APA
Wolfram Language. (2010). AlternatingGroup. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AlternatingGroup.html
BibTeX
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