How Does Conjugation Affect Cycles in Permutations?

[ForMyThunder]

Homework Statement

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

Homework Equations

N/A

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 = (i_m1i_m2...i_mk) be a cycle and let D = (i₁i₂...i_r). Then,

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

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

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

 

[tiny-tim]

Hi ForMyThunder!

Hint: what is P^-1(P(i₁)) ?