### Conjugacy Classes

The conjugacy class of * pÎS*_{n} will be denoted by *C*^{S}( p),
the centralizer by *C*_{S}( p), so that we obtain the following descriptions
and properties of conjugacy classes and centralizers of elements of * S*_{n} :

**Corollary: **
*
Let ** p* and * s* denote elements of * S*_{n} . Then
*C*^{S}( p)=C^{S}( s) iff a( p)= a( s) iff a( p)=a( s).
*C*^{S}( p)=C^{S}( p^{-1}), i.e. * S*_{n} is *ambivalent*
, which means that each element is a conjugate of its
inverse.
* | C*_{S}( p) | = Õ_{i} i^{ai( p)}a_{i}( p)!, and
| C^{S}( p) | = n!/ Õ_{i} i^{ai( p)}a_{i}( p)!.
There are some examples to compute the orders of the
conjugacy classes and centralizers
in * S*_{n} .
* | ápñ | = lcm { a*_{i}( p) | i Î __c( p)__ }= lcm {i | a_{i}( p)>0 }.
- Each proper partition
* a|¾n* occurs as the cycle
partition
of some * pÎS*_{n} .

(The first, second, fourth and fifth item is clear from the foregoing, while the
third one follows from the fact that there are exactly *i*^{ai}a_{i}! mappings
which map a set of *a*_{i} *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 * S*_{n} (and correspondingly in * S*_{n} )
by partitions or cycle types putting

*C*^{a} := C^{a} := C^{S}( p),
when a( p)= a, and a( p)
=a.

last changed: January 19, 2005