Dihedral Group (Subscript[D, 4])

Author

Eric W. Weisstein
September 26, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/DihedralGroupD4.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DihedralGroupD4.html.

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

Representation

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

Permutation Group

pg=DihedralGroup[4]

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

GroupQ[pg]

Matrix Group

MatrixForm/@(mg=DihedralGroupMatrices[4])//TraditionalForm

mrules=Thread[mg->letters];

GroupQ[mg]

Properties

Nonabelian

AbelianGroupQ/@{mg,pg}

Cycle Index

DihedralGroupIndex[4,x]/.x[n_]:>Subscript[x,n]//TraditionalForm

Conjugacy Classes

conj=ConjugacyClasses[mg]/.mrules

conj=ConjugacyClasses[pg]/.prules

Length/@conj

Length[%]

Subgroups

(subg=Subgroups[pg])/.prules

Length[subg]

Through[{Length,First}[#]]&/@Split[Length/@subg]

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

Multiplication Table

GroupMultiplicationTable[pg,prules]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[pg,nrules][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[(Subscript[D, 4])],
DisplayFunction->Identity]&/@{Hue[#]&,GrayLevel}]]

Powers

Table[MatrixPower[#,n]&/@mg,{n,8}]/.mrules//TableForm

Count[#,"1"]&/@(Table[MatrixPower[#,n]&/@mg,{n,8}]/.mrules)

Cycle Graph

ShowLabeledGraph[RadialEmbedding[CycleGraph[pg]]]