Dihedral Group (Subscript[D, 5])

Author

Eric W. Weisstein
September 26, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/DihedralGroupD5.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DihedralGroupD5.html.

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

Representation

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

Permutation Group

pg=DihedralGroup[5]

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

GroupQ[pg]

Matrix Group

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

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

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

Properties

Non-Abelian

AbelianGroupQ[pg]

Cycle Index

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

Conjugacy Classes

Sort/@(conj=ConjugacyClasses[pg]/.prules)

Length[conj]

Length/@conj

Subgroups

Length[subg=Subgroups[pg]]

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

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[D, 5])],
DisplayFunction->Identity]&/@{Hue[#]&,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

ShowLabeledGraph[RadialEmbedding[CycleGraph[DihedralGroup[5]]]]