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 a|¾n 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.
last changed: January 19, 2005