Cyclic Group (Subscript[C, 2])

Author

Eric W. Weisstein
September 26, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/CyclicGroupC2.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CyclicGroupC2.html.

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

Representations

letters={"1","A"};

Modulo Multiplication Group

Position[CharacteristicFactors[ModuloMultiplicationPhi[#]]&/@Range[10000],{2}]//Flatten//Timing

ModuloMultiplicationPhi/@{3,4,6}

PermutationGroup

pg=CyclicGroup[2]

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

GroupQ[pg]

Matrix Group

MatrixForm/@(mg=CyclicGroupMatrices[2])//TraditionalForm

mrules=Thread[mg->letters];

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

Properties

Abelian

AbelianGroupQ/@{pg,mg}

Cycle Index

CyclicGroupIndex[2,x]/.x[n_]:>Subscript[x,n]//TraditionalForm

Subgroups

Length[subg=Subgroups[pg]]

Subgroups[pg]/.prules

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

Conjugacy Classes

Length[ConjugacyClasses[pg]]

ConjugacyClasses[pg]/.prules

Multiplication Table

GroupMultiplicationTable[pg,prules]

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

Powers

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[2]]],AspectRatio->Automatic,PlotLabel->TraditionalForm[(Subscript[C, 2])],
ImageSize->150,
TextStyle->{FontFamily->"Times",FontSize->20}]

ShowLabeledGraph[RadialEmbedding[CycleGraph[CyclicGroup[2]]]]