Modulo Multiplication Group

Author

Eric W. Weisstein
August 29, 2005

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

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

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

Typesetting

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

Example

Grid[MapIndexed[{Subscript[M,#2[[1]]],Sequence@@#1}&,Table[{EulerPhi[n],MyTimes[Subscript[C,#]&/@CharacteristicFactors[ModuloMultiplicationPhi[n]]],First/@ModuloMultiplicationGroup[n]},{n,2,24}]],GridFrame->True,
ColumnLines->True,ColumnAlignments->{Left,Right,Left,Left}]/.MyTimes[{}]:>⟨e⟩//TraditionalForm

Cycle Graphs

{#,GroupCycles[#]}&/@Range[2,16]//ColumnForm

Off[General::"stop"]

Show[GraphicsArray[
Partition[ShowLabeledGraph[RecognizeGraph[CycleGraph[ModuloMultiplicationGroup[#]]],
First/@ModuloMultiplicationGroup[#],
DisplayFunction->Identity,PlotRange->All,
AspectRatio->Automatic,
PlotLabel->TraditionalForm[Subscript[M,#]],VertexStyle->Disk[Large],VertexColor->Red]&/@Range[2,16],5]]]

Show[GraphicsArray[
Partition[ShowLabeledGraph[ModuloMultiplicationGraph[#],DisplayFunction->Identity,PlotRange->All,
AspectRatio->Automatic,
PlotLabel->TraditionalForm[Subscript[M,#]],VertexStyle->Disk[Large],VertexColor->Red]&/@Prime/@Range[3,8],3]]]

As a Function of Order

Table[{n,MyTimes[Subscript[C,#]&/@CharacteristicFactors[ModuloMultiplicationPhi[n]]]//TraditionalForm},{n,3,24}]//ColumnForm

Cyclic

Position[Length/@CharacteristicFactors/@ModuloMultiplicationPhi/@Range[30],1]//Flatten

Orders for Which Elements are Self-Conjugate

Position[Table[And@@SameQ@@@({#,GroupInverse[#]}&/@ModuloMultiplicationGroup[n]),{n,500}],True]//Flatten

Direct Products of (Subscript[C, 2])

Position[ModuloMultiplicationPhi/@Range[1000],{2}]//Flatten

Position[ModuloMultiplicationPhi/@Range[1000],{2,2}]//Flatten

Position[ModuloMultiplicationPhi/@Range[1000],{2,2,2}]//Flatten

Position[ModuloMultiplicationPhi/@Range[10000],{2,2,2,2}]//Flatten

Position[ModuloMultiplicationPhi/@Range[10000],{2,2,2,2,2}]//Flatten

Orders Having a Given Group

l=Sort[Table[{CharacteristicFactors[ModuloMultiplicationPhi[n]],n},{n,3,50}],
If[Unequal@@(t=Times@@@First/@{##}),Less@@t,Less@@Length/@First/@{##}]&]

{MyTimes[Subscript[C,#]&/@#[[1,1]]]//TraditionalForm,Sort[Last/@#]}&/@Split[l,(SameQ@@First/@{##}&)]//ColumnForm

m={CharacteristicFactors[ModuloMultiplicationPhi[#]],#}&/@Range[5000];//Timing

Sort[Select[{#[[1,1]],Last/@#}&/@Split[Sort[m],First[#1]==First[#2]&],Length[Last[#]]>10&],Length[Last[#1]]>Length[Last[#2]]&]//ColumnForm