Conjugacy Classes

### Conjugacy Classes

The conjugacy class of pÎSn will be denoted by CS( p), the centralizer by CS( p), so that we obtain the following descriptions and properties of conjugacy classes and centralizers of elements of Sn :

Corollary: Let p and s denote elements of Sn . Then
• CS( p)=CS( s) iff a( p)= a( s) iff a( p)=a( s).
• CS( p)=CS( p-1), i.e. Sn is ambivalent , which means that each element is a conjugate of its inverse.
• | CS( p) | = Õi iai( p)ai( p)!, and | CS( p) | = n!/ Õi iai( p)ai( p)!. There are some examples to compute the orders of the conjugacy classes and centralizers in Sn .
• | ápñ | = lcm { ai( p) | i Î c( p) }= lcm {i | ai( p)>0 }.
• Each proper partition an occurs as the cycle partition of some pÎSn .

(The first, second, fourth and fifth item is clear from the foregoing, while the third one follows from the fact that there are exactly iaiai! mappings which map a set of ai i-tuples onto this same set up to cyclic permutations inside each i-tuple.) For the sake of simplicity we can therefore parametrize the conjugacy classes of elements in Sn (and correspondingly in Sn ) by partitions or cycle types putting

Ca := Ca := CS( p), when a( p)= a, and a( p) =a.

 harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/ UNI-Graz Institut für Mathematik UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005

 Conjugacy Classes