Pair Group

Author

Eric W. Weisstein
March 11, 2002

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

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

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

Initialization

<<DiscreteMath`Combinatorica`

?PairGroupIndex

SymmetricGroup[3]

PairGroupIndex[SymmetricGroup[3],x]

?PairGroup

Examples

Lengths

Table[PairGroup[SymmetricGroup[n]][[1]]//Length,{n,5}]

Table[Binomial[n,2],{n,5}]

p=2

SymmetricGroup[2]

PairGroup[SymmetricGroup[2]]

PairGroup[AlternatingGroup[2]]

p=3

SymmetricGroup[3]

PairGroup[SymmetricGroup[3]]

AlternatingGroup[3]

PairGroup[AlternatingGroup[3]]

p=3

SymmetricGroup[4]

PairGroup[SymmetricGroup[4]]

Cycle Index

Table[CycleIndex[PairGroup[SymmetricGroup[n]],x],
{n,6}]/.x[n_]:>Subscript[x,n]//ColumnForm//TraditionalForm

Table[Expand[n! CycleIndex[PairGroup[SymmetricGroup[n]],x]],
{n,6}]/.x[n_]:>Subscript[x,n]//ColumnForm//TraditionalForm