Cyclic Group (Subscript[C, 5])

Author

Eric W. Weisstein
September 26, 2009

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GroupTheory/CyclicGroupC5.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/CyclicGroupC5.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Representation

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

Modulo Multiplication Group

Position[CharacteristicFactors[ModuloMultiplicationPhi[#]]&/@Range[10000],{5}]//Flatten//Timing

Permutation Group

pg=CyclicGroup[5]

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

GroupQ[pg]

Matrix Group

MatrixForm/@(mg=CyclicGroupMatrices[5])//TraditionalForm

mrules=RootReduce[Thread[mg->letters]];

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

Properties

Abelian

Map[RootReduce,AbelianGroupQ/@{pg,mg},{2}]

Cycle Index

CyclicGroupIndex[5,x]/.x[n_]:>Subscript[x,n]//TraditionalForm

Conjugacy Classes

Length[ConjugacyClasses[pg]]

Sort[ConjugacyClasses[pg]/.prules]

Subgroups

Length[subg=Subgroups[pg]]

subg/.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, 5])],
DisplayFunction->Identity]&/@{Hue[#/2]&,GrayLevel}]]

Powers

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

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

Cycle Graph

Show[Graphics[CycleDiagram[CyclicGroup[5]]],AspectRatio->Automatic,PlotLabel->TraditionalForm[(Subscript[C, 5])],
ImageSize->200,
TextStyle->{FontFamily->"Times",FontSize->20}]

ShowLabeledGraph[CycleGraph[CyclicGroup[5]]]