Symmetric Group

Author

Eric W. Weisstein
October 6, 2003

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

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

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

Initialization

<<Groups`

Usage

?SymmetricGroup

Transitivity

TransitiveGroupQ/@Table[SymmetricGroup[n],{n,2,6}]

Group Tables

SymmetricGroupTable[n_]:=Module[{p=Permutations[Range[n]],i,j},
Table[p[[i]][[p[[j]]]],{i,n!},{j,n!}]
]

SymmetricGroup[2]

SymmetricGroup[3]

SymmetricGroupTable[3]

m=Map[FromDigits,SymmetricGroupTable[2],{2}];
TableForm[m,
TableHeadings->{#,#}&[m[[1]]]]

m=Map[FromDigits,SymmetricGroupTable[3],{2}];
TableForm[m,
TableHeadings->{#,#}&[m[[1]]]]

Cycle Index

Implementation

<<PartitionsJ`;
<<DiscreteMath`Combinatorica`;

SymmetricZ[p_,a_]:=Module[
{
ar=Array[a,p]
},
Plus@@(Times@@(ar^#)/ Product[k^(#[[k]])#[[k]]!,{k,p}]&/@PartitionsJ[p])
]

Table[SymmetricZ[n,x] -CycleIndex[SymmetricGroup[n],x],
{n,2,6}]

Timing

v5.0

SymmetricZ[30,x];//Timing

SymmetricGroupIndex[30,x];//Timing

v4.2

CycleIndex[SymmetricGroup[7],x];//Timing

SymmetricGroupIndex[7,x];//Timing

Table of values

SymmetricGroupIndex[5,x]

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

Multiplication Table

g=SymmetricGroup[4];
nrules=Thread[g->Range[Length[g]]];
Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[g,nrules][[1]],ColorFunction->#,Frame->False,Mesh->True,PlotLabel->TraditionalForm[(Subscript[S, 4])],
DisplayFunction->Identity]&/@{Hue,GrayLevel}]]