WOLFRAM

Wolfram Language & System Documentation Center

GroupCentralizer[group,g]

returns the centralizer of the element g in group.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
See Also
Tech Notes
Related Guides
History
Cite this Page

GroupCentralizer

GroupCentralizer[group,g]

returns the centralizer of the element g in group.

Details

  • The centralizer of an element g in group is the subgroup of elements of group that commute with g.

Examples

open all close all

Basic Examples  (1)

Take a permutation group:

Wolfram Language code: GroupOrder[group = PermutationGroup[{Cycles[{{1, 2, 7}, {3, 10, 5, 4}}], Cycles[{{1, 4, 2, 5, 3}}]}]]

And a permutation:

Wolfram Language code: g = Cycles[{{3, 7}, {4, 5}, {6, 8, 10, 9}}];

This is the subgroup of permutations in the group that commutes with the permutation g:

Wolfram Language code: GroupCentralizer[group, g]
Wolfram Language code: GroupOrder[%]

Scope  (1)

Take a permutation group:

Wolfram Language code: GroupOrder[group = PermutationGroup[{Cycles[{{1, 7, 2}, {3, 10, 5, 4}, {6, 8, 9}}], Cycles[{{1, 4, 2, 5, 3}}]}]]

And a permutation:

Wolfram Language code: g = Cycles[{{3, 7}, {4, 5}, {6, 8, 10, 9}}];

This is the subgroup of permutations in the group that commutes with the permutation g:

Wolfram Language code: centralizer = GroupCentralizer[group, g]
Wolfram Language code: GroupOrder[%]

Check the result by direct computation of products with all elements:

Wolfram Language code: centralizerlist = GroupElements[centralizer]

This function checks that a permutation h commutes with g:

Wolfram Language code: gcomQ[h_] := PermutationProduct[h, g] === PermutationProduct[g, h]

All 16 permutations in the centralizer commute with g:

Wolfram Language code: gcomQ /@ centralizerlist
Wolfram Research (2010), GroupCentralizer, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupCentralizer.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_groupcentralizer, author="Wolfram Research", title="{GroupCentralizer}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GroupCentralizer.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_groupcentralizer, organization={Wolfram Research}, title={GroupCentralizer}, year={2010}, url={https://reference.wolfram.com/language/ref/GroupCentralizer.html}, note=[Accessed: 13-June-2026]}