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RightCosetRepresentative[group,g]

returns the smallest element in the right coset of products of the elements of group by g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
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RightCosetRepresentative

RightCosetRepresentative[group,g]

returns the smallest element in the right coset of products of the elements of group by g.

Details and Options

  • The representative is chosen as the smallest element in the coset, as determined by the ordering given by Less.

Examples

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Basic Examples  (2)

Representative of a coset:

Wolfram Language code: RightCosetRepresentative[PermutationGroup[{Cycles[{{1, 7}}], Cycles[{{1, 9, 7}, {2, 4}, {3, 5}}]}], Cycles[{{2, 9, 8}}]]

For an element in the group, the representative is the identity:

Wolfram Language code: RightCosetRepresentative[PermutationGroup[{Cycles[{{1, 7}}], Cycles[{{1, 9, 7}, {2, 4}, {3, 5}}]}], Cycles[{{2, 4}, {3, 5}, {7, 9}}]]

Scope  (1)

Compute the canonical representative of a right coset:

Wolfram Language code: RightCosetRepresentative[PermutationGroup[{Cycles[{{1, 7}}], Cycles[{{1, 9, 7}, {2, 4}, {3, 5}}]}], Cycles[{{2, 9, 8}}]]

List all elements of the corresponding right coset:

Wolfram Language code: PermutationProduct[#, Cycles[{{2, 9, 8}}]]& /@ GroupElements[PermutationGroup[{Cycles[{{1, 7}}], Cycles[{{1, 9, 7}, {2, 4}, {3, 5}}]}]]

The smallest element is the canonical representative:

Wolfram Language code: First@Sort[%, Less]

Properties & Relations  (1)

The inverse of the canonical representative of a right coset belongs to the left coset but is not canonical in general. Take a permutation and a group:

Wolfram Language code: g = Cycles[{{2, 9, 8}}]; group = PermutationGroup[{Cycles[{{1, 7}}], Cycles[{{1, 9, 7}, {2, 4}, {3, 5}}]}];

This is the left coset of the inverse of that element:

Wolfram Language code: PermutationProduct[InversePermutation@g, #]& /@ GroupElements[group]

The inverse of the canonical right coset representative belongs to the coset but is not the canonical representative:

Wolfram Language code: InversePermutation@RightCosetRepresentative[group, g]
Wolfram Language code: First@Sort[%%, Less]
Wolfram Research (2010), RightCosetRepresentative, Wolfram Language function, https://reference.wolfram.com/language/ref/RightCosetRepresentative.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_rightcosetrepresentative, author="Wolfram Research", title="{RightCosetRepresentative}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/RightCosetRepresentative.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_rightcosetrepresentative, organization={Wolfram Research}, title={RightCosetRepresentative}, year={2010}, url={https://reference.wolfram.com/language/ref/RightCosetRepresentative.html}, note=[Accessed: 13-June-2026]}