Cyclic Group

Author

Eric W. Weisstein
December 9, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Notation

CharacteristicFactors[l_]:=Module[{ll=Split[Sort[Flatten[FactorInteger/@l,1]],Equal@@First/@{##}&]},
Reverse[Times@@@Apply[Power,Map[Last,NestList[DeleteCases[Map[Drop[#,-1]&,#,{1}],{}]&,ll,Max[Length/@ll]-1],{2}],{2}]]
]

MakeBoxes[MyTimes[l_List],TraditionalForm]:=RowBox[Drop[Flatten[{MakeBoxes[#,TraditionalForm],"×"}&/@l],-1]]

Cycle Index

Implementation

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

Expand[Table[CyclicZ[n,x ] -CyclicGroupIndex[n,x],
{n,2,6}]]

Timing

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

CyclicGroupIndex[30,x];//Timing

Table of values

CyclicGroupIndex[5,x]

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

Multiplication Table

V5

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

V7

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