# Homework Help: Permutation and cycles

1. Dec 29, 2009

### ForMyThunder

1. The problem statement, all variables and given/known data

Let P be a permutation of a set. Show that P(i1i2...ir)P-1 = (P(i1)P(i2)...P(ir))

2. Relevant equations

N/A

3. The attempt at a solution

Since P is a permutation, it can be written as the product of cycles. So I figured that showing that the above equation holds for cycles will be sufficient to show that it holds for all permutations.

Let C = (im1im2...imk) be a cycle and let D = (i1i2...ir). Then,

imk$$\stackrel{C^{-1}}{\rightarrow}$$imk-1$$\stackrel{D}{\rightarrow}$$imk-1+1$$\stackrel{C}{\rightarrow}$$imk+1

Let D` = (C(i1)C(i2)...C(ir)), then imk$$\stackrel{}{\rightarrow}$$imk+1

I don't know how to prove this last part, nor do I know if my reasoning is correct. Any suggestions?

Last edited: Dec 29, 2009
2. Dec 29, 2009

### tiny-tim

Hi ForMyThunder!

Hint: what is P-1(P(i1)) ?