Cyclic Group (Subscript[C, 3])

Author

Eric W. Weisstein
October 6, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/CyclicGroupC3.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CyclicGroupC3.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Representation

letters=Prepend[CharacterRange["A","B"],"1"];

Modulo Multiplication Group

Position[CharacteristicFactors[ModuloMultiplicationPhi[#]]&/@Range[10000],{3}]//Flatten//Timing

PermutationGroup

pg=CyclicGroup[3]

prules=Thread[pg->letters];
nrules=Thread[pg->Range[Length[pg]]];

GroupQ[pg]

Matrix Group

MatrixForm/@(mg=CyclicGroupMatrices[3])//TraditionalForm

mrules=Thread[mg->letters];

GroupQ[mg,Simplification->(RootReduce[FullSimplify[#]]&)]//Timing

Properties

Abelian

AbelianGroupQ/@{pg,mg}

Cycle Index

CyclicGroupIndex[3,x]/.x[n_]:>Subscript[x,n]//TraditionalForm

Conjugacy Classes

Length[ConjugacyClasses[pg]]

Sort[ConjugacyClasses[pg]/.prules]

Subgroups

Length[subg=Subgroups[pg]]

Subgroups[pg]/.prules

Select[subg,NormalSubgroupQ[#,pg]&]/.prules

Multiplication Table

GroupMultiplicationTable[pg,prules]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[pg,nrules][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[(Subscript[C, 3])],
DisplayFunction->Identity]&/@{Hue[#/2]&,GrayLevel}]]

Powers

Table[MatrixPower[#,n]&/@mg,{n,3}]/.mrules//TableForm

Count[#,"1"]&/@(Table[GroupTimes[Sequence@@Table[#,{n}]]&/@pg,{n,Length[pg]}]/.prules)

Cycle Graph

Show[Graphics[CycleDiagram[CyclicGroup[3]]],AspectRatio->Automatic,PlotLabel->TraditionalForm[(Subscript[C, 3])],
ImageSize->200,
TextStyle->{FontFamily->"Times",FontSize->20}]

ShowLabeledGraph[CycleGraph[CyclicGroup[3]]]