Normalizers, Centralizers and orbits

From: Warren065 (warren065_at_aol.com)
Date: 11/20/04 

Date: 20 Nov 2004 08:38:40 GMT

I have the following problem:

Let P be a subgroup of S_n (the symmetric group) where P is of prime order and
suppose x belongs to S_n normalizes but DOES NOT centralize P. Show that x
fixes at most one point in each orbit of P.

[My proof is more of a wordy explanation. I just can't think of how to prove
this explicitly]

Proof: Let P be a subgroup of S_n of prime order. So P=(1,.....,p). Suppose x
belongs to S_n fixes 1. Now if x fixes any other point in (1,....,p) and
normalizes (1,2,....,p) then it fixes all points 1,2,....,p. Then x would be
an element of the centralizer. So if x fixes 2 points in an orbit of P, then
it centralizes P. Therefore x fixes at most one piont in each orbit of P.