Dihedral Group

Author

Eric W. Weisstein
October 6, 2003

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

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

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

Initialization

<<Groups`

Usage

?DihedralGroup

Dihedral Groups

Length/@DihedralGroup/@Range[5]

Therefore, n=1 and 2 don't actually give dihedral groups

Abelian

AbelianGroupQ/@DihedralGroup/@Range[3,10]

Cycle Index

Implementation

CyclicZ[p_,a_]:=Module[{},
Plus@@(EulerPhi[#]a[#]^(p/#)&/@Divisors[p])/p
]

DihedralZ[p_,a_]:=Module[{},
CyclicZ[p,a]/2+Switch[p,
_?OddQ,a[1]a[2]^((p-1)/2)/2,
_?EvenQ,(a[2]^(p/2)+a[1]^2a[2]^((p-2)/2))/4
]
]

Expand[Table[DihedralZ[n,x ] -DihedralGroupIndex[n,x],
{n,2,6}]]

Timing

CycleIndex[DihedralGroup[30],x];//Timing

DihedralGroupIndex[30,x];//Timing

Table of values

DihedralGroupIndex[5,x]

Table[DihedralGroupIndex[n,x]/.x[n_]:>Subscript[x,n],{n,5}]//ColumnForm//TraditionalForm

Multiplication Table

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[DihedralGroup[10],Thread[DihedralGroup[10]->Range[20]]][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[(Subscript[D, 10])],
DisplayFunction->Identity]&/@{Hue,GrayLevel}]]