Conjugacy Classes |

Corollary:Letpandsdenote elements ofS. Then_{n}

C.^{S}( p)=C^{S}( s) iff a( p)= a( s) iff a( p)=a( s)C, i.e.^{S}( p)=C^{S}( p^{-1})Sis_{n}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 Îc( p)}= lcm {i | a_{i}( p)>0 }- Each proper partition
a|¾noccurs as the cycle partition of somepÎS._{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

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

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Conjugacy Classes |