TITLE: What is an intuition behind conjugate permutations?
QUESTION [10 upvotes]: I know the definition of conjugate permutations.
 $$\exists p \quad p^{-1} \alpha p=\beta$$
So the $\alpha$ and $\beta$ is a pair of conjugate permutations. But can anybody can give some concise, vivid example to describe conjugate permutations? To help me understand the relation of $\alpha$ and $\beta$ intuitively? I'm a newbie in group-theory.
First time here. If there is anything unsuitable in my specification, do tell me please.

REPLY [8 votes]: A little more explicitly:
Let's say you have the permutation $\alpha$ with $\alpha(1) = 3$, $\alpha(2) = 1$, $\alpha(3) = 2$.  Let's say your are afraid of numbers, so to make this permutation friendly you use letters instead.  You relabel $1$ as $a$, $2$ as $b$, $3$ as $c$ to get a new permutation $\beta$ of the letters $\{a,b,c\}$.   We get $\beta(a) = c$, $\beta(b) = a$, $\beta(c) = b$.
To figure out more carefully what is going on, define $p:\{a,b,c\} \rightarrow \{1,2,3\}$ by $p(a) = 1,p(b) = 2, p(c) = 3$, so that $p$ is the "translator" between your sets of symbols.  You want to know where $\alpha$ should do in this translated alphabet.  To find out what should happen to $a$, first translate: $p(a) = 1$.  Then apply $\alpha$: $\alpha(1) = 3$.  Finally, translate back using $p^{-1}$: $p^{-1}(3) = c$.  So the "translated" version of $\alpha$ is the map $\beta = p^{-1} \alpha p$ with $\beta(a) = c$.  W get $$\beta(a) = p^{-1}\alpha p(a) = p^{-1} \alpha(1) = p^{-1}(3) = c,$$  $$\beta(b) = p^{-1} \alpha p(b) = p^{-1} \alpha(2) = p^{-1} (1) = a,$$ $$\beta(c) = p^{-1} \alpha p(c) = p^{-1} \alpha (3) =  p^{-1} (2) = b$$ which is what we want.
Here we're using two sets of symbols, $\{1,2,3\}$ and $\{a,b,c\}$, but we might as well relabel using the same set of symbols $\{1,2,3\}$ and $\{1,2,3\}$.  The same idea works, and then you have the notion of conjugacy in $S_n$: if $\beta = p^{-1} \alpha p$, then $\beta$ is just given by taking $\alpha$ and relabeling the symbols using $p$ as a "translation".