({{, (RefLink[GroupCentralizer, paclet:ref/GroupCentralizer])[group,g] 
returns the centralizer of the element g in group.}})

OpenersNotesSection(Spacer1)Details

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

OpenersPrimaryExamplesSection(Spacer1)Examples  (2)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

Take a permutation group:

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

And a permutation:

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:

GroupCentralizer[group,g]

GroupOrder[%]

OpenersExampleSection1(Spacer1)Scope  (1)

Take a permutation group:

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

And a permutation:

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:

centralizer=GroupCentralizer[group,g]

GroupOrder[%]

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

centralizerlist=GroupElements[centralizer]

This function checks that a permutation h commutes with g:

gcomQ[h_]:=PermutationProduct[h,g]===PermutationProduct[g,h]

All 16 permutations in the centralizer commute with g:

gcomQ/@centralizerlist

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Cycles, paclet:ref/Cycles])  ▪  (RefLink[PermutationProduct, paclet:ref/PermutationProduct])  ▪  (RefLink[PermutationGroup, paclet:ref/PermutationGroup])  ▪  (RefLink[GroupElements, paclet:ref/GroupElements])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Permutation Groups, paclet:tutorial/PermutationGroups])}, {▪(RefLinkPlain[Group Theory Algorithms, paclet:tutorial/GroupTheoryAlgorithms])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Group Theory, paclet:guide/GroupTheory])}})}})