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PermutationSupport[perm]

returns the support of the permutation perm.

Details
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Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Properties & Relations  
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PermutationSupport

PermutationSupport[perm]

returns the support of the permutation perm.

Details

  • The support of a permutation perm is the list of integers that are not fixed by perm.
  • The list of integers is returned sorted.
  • PermutationSupport works with Cycles objects as well as with permutation lists. When applied to a permutation list, PermutationSupport[{p1,…,pn}] returns the pi for which pi≠i.

Examples

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Basic Examples  (2)

Points moved by a permutation in disjoint cyclic form:

Wolfram Language code: PermutationSupport[Cycles[{{1, 7}, {2, 5, 10, 9}, {4, 6}}]]

Points moved in a permutation list:

Wolfram Language code: PermutationSupport[{1, 4, 3, 5, 2, 6, 10, 7, 9, 8}]

Scope  (2)

Support of permutations in disjoint cyclic form:

Wolfram Language code: PermutationSupport[Cycles[{{1, 8, 5}, {2, 9, 4, 3}}]]
Wolfram Language code: PermutationSupport[Cycles[{}]]

Support of permutation lists:

Wolfram Language code: PermutationSupport[{5, 3, 4, 9, 8, 6, 7, 1, 2}]
Wolfram Language code: PermutationSupport[{1, 2, 3, 4, 5, 6, 7, 8, 9}]
Wolfram Language code: PermutationSupport[{}]

Generalizations & Extensions  (1)

The support of a permutation group is defined as the union of the supports of its elements:

Wolfram Language code: PermutationSupport[PermutationGroup[{Cycles[{{1, 3, 5, 7}}], Cycles[{{1, 2}, {3, 4}}]}]]

Support of the default permutation representation of a named abstract group:

Wolfram Language code: PermutationSupport[DihedralGroup[5]]

Properties & Relations  (2)

The support of a permutation is returned as an ordered list of integers:

Wolfram Language code: PermutationSupport[Cycles[{{2, 6}, {1, 5, 3, 17}}]]
Wolfram Language code: OrderedQ[%]

The permutation support of a permutation in cyclic form is the union of its cycles:

Wolfram Language code: PermutationSupport[Cycles[{{1, 7}, {2, 5, 10, 9}, {4, 6}}]] === Union@@First[Cycles[{{1, 7}, {2, 5, 10, 9}, {4, 6}}]]
Wolfram Research (2010), PermutationSupport, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationSupport.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_permutationsupport, author="Wolfram Research", title="{PermutationSupport}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationSupport.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_permutationsupport, organization={Wolfram Research}, title={PermutationSupport}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationSupport.html}, note=[Accessed: 12-June-2026]}