Alternating Group

Author

Eric W. Weisstein
October 6, 2003

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

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

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

Initialization

<<Groups`

Usage

?AlternatingGroup

Length

Length/@AlternatingGroup/@Range[2,9]

Transitivity

TransitiveGroupQ/@AlternatingGroup/@Range[2,7]//Timing

TransitiveGroupQ[AlternatingGroup[8]]//Timing

TransitiveGroupQ[AlternatingGroup[9]]//Timing

Abelian

AbelianGroupQ/@AlternatingGroup/@Range[2,6]//Timing

AbelianGroupQ[AlternatingGroup[7]]//Timing

Cycle Index

Code

<<DiscreteMath`Combinatorica`;
<<PartitionJ`;

AlternatingZ[p_,a_]:=Module[
{
ar=Array[a,p]
},
Plus@@(Times@@(ar^#)(1+(-1)^(Plus@@Take[#,{2,p,2}]))/ Product[k^(#[[k]])#[[k]]!,{k,p}]&/@PartitionsJ[p])
]

Comparison with Combinatorica

Table[AlternatingZ[n,x ] -AlternatingGroupIndex[n,x],
{n,2,10}]//Timing

Timing

AlternatingZ[30,x];//Timing

AlternatingGroupIndex[30,x];//Timing

Values

AlternatingGroupIndex[5,x]

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

Conjuacy Classes

Length/@ConjugacyClasses/@AlternatingGroup/@Range[2,6]//Timing

Length[ConjugacyClasses[AlternatingGroup[7]]]//Timing

ConjugacyClasses[AlternatingGroup[4]]/.Thread[AlternatingGroup[4]->Range[12]]

Length/@%

Subgroups

Length/@Subgroups/@AlternatingGroup/@Range[2,4]//Timing

Length[Subgroups[AlternatingGroup[5]]]//Timing

Through[{Length,First}[#]]&/@Split[Length/@(subg=Subgroups[AlternatingGroup[4]])]

Length[subg]

Length/@Subgroups/@AlternatingGroup/@Range[2,4]//Timing

Length[Select[subg,NormalSubgroupQ[#,AlternatingGroup[4]]&]]

Normal Subgroups

Function[g,Count[Subgroups[g],_?(NormalSubgroupQ[#,g]&)]]/@AlternatingGroup/@Range[2,4]//Timing

Take[Select[Subgroups[AlternatingGroup[4]],NormalSubgroupQ[#,AlternatingGroup[4]]&],{2,-2}]//Timing

Cycle Graph

MaximalGraphCycles[AlternatingGroup[4]]

ShowLabeledGraph[RadialEmbedding[CycleGraph[AlternatingGroup[4]]]]

Multiplication Table

g=AlternatingGroup[5];
nrules=Thread[g->Range[Length[g]]];
Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[g,nrules][[1]],ColorFunction->#,Frame->False,Mesh->True,PlotLabel->TraditionalForm[(Subscript[A, 5])],
DisplayFunction->Identity]&/@{Hue,GrayLevel}]]

Powers

Count[#,"1"]&/@(Table[GroupTimes[Sequence@@Table[#,{n}]]&/@AlternatingGroup[4],{n,12}]/.Thread[AlternatingGroup[4]->Prepend[CharacterRange["A","K"],"1"]])