Dihedral Group (Subscript[D, 3])

Author

Eric W. Weisstein
September 26, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/DihedralGroupD3.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DihedralGroupD3.html.

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

Representation

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

Permutation Group

pg=DihedralGroup[3]

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

GroupQ[pg]

Matrix Matrix

MatrixForm/@(mg0=DihedralGroupMatrices[3])//TraditionalForm

MatrixForm/@(mg=mg0[[{1,2,3,5,4,6}]])//TraditionalForm

mrules=Thread[mg->letters];

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

Flatten[Outer[Times,S/@{0,1},R/@Range[0,2],1],1]/.x_[n_]:>Superscript[x,n]//TraditionalForm

1-D

Det/@mg

Properties

Abelian

AbelianGroupQ/@{pg,mg}

Cycle Index

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

Conjugacy Classes

Length[ConjugacyClasses[pg]]

ConjugacyClasses[pg]/.prules

ConjugacyClasses[mg]/.mrules

Map[TraditionalForm,ConjugacyClassTable[pg/.prules],{2}]//TableForm

Subgroups

Length[Subgroups[pg]]

Subgroups[pg]/.prules

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

Multiplication Table

GroupMultiplicationTable[pg,prules]

GroupMultiplicationTable[mg,mrules]

GroupMultiplicationTable[mg,mrules]==GroupMultiplicationTable[pg,prules]

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

Powers

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

(Table[GroupTimes[Sequence@@Table[#,{n}]]&/@mg,{n,Length[mg]}]/.mrules)//TableForm

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

Cycle Graph

ShowLabeledGraph[RadialEmbedding[CycleGraph[DihedralGroup[3]]]]