Conjugacy Classes

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^{a_i( p)}a_i( p)!, and | C^S( p) | = n!/ Õ_i i^{a_i( 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^a_ia_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.

Conjugacy Classes