Finite Group (Subscript[C, 2])×(Subscript[C, 2])

Author

Eric W. Weisstein
September 13, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/FiniteGroupC2C2.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FiniteGroupC2C2.html.

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

Initialization

<<Groups`

Representation

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

Modulo Multiplication Group

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

ModuloMultiplicationPhi/@{8,12}

ModuloMultiplicationGroup/@{8,12}

Matrix Group

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

rules=Thread[mg->letters];
nrules=Thread[mg->Range[Length[mg]]];

GroupQ[mg]

Properties

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

AbelianGroupQ[g]

Multiplication Table

GroupMultiplicationTable[mg,rules]

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

Powers

Table[MatrixPower[#,n]&/@g,{n,4}]/.rules//TableForm

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

Conjugacy Classes

Map[TraditionalForm,ConjugacyClassTable[g/.rules],{2}]//TableForm

ConjugacyClasses[g]/.rules

Subgroups

Subgroups[g]/.rules

Select[Subgroups[g],NormalSubgroupQ[#,g]&]/.rules

Cycle Graph

GroupCycles[g]/.rules

ShowLabeledGraph[CycleGraph[g]]