Doing an assignment, getting a bit frustrated with this exercise, would really appreciate some help. The first exercise explains what $\sigma$ and $\rho$ are:
Let $\sigma$ be the $r$-cycle $(1,2,3,\dots,r)$ in $S_n$. Show that $\sigma$ is conjugate to its own inverse; that is, there is a permutation such that $\rho \sigma \rho^{-1}=\sigma^{-1}$
A permutation that reverses the order of $1234\dots r$ does this, e.g. $\rho=\left(1,r\right)\left(2,r-1\right)\left(3,r-2\right)\dots\left(r-2,3\right) \left(r-1,2\right)$, then:
$\rho\sigma\rho^{-1} = \left(\begin{array}{cccccccc} 1 & 2 & 3 & 4 & \dots & r-2 & r-1 & r\\ r & r-1 & r-2 & r-3 & \dots & 3 & 2 & 1\\ 1 & r & r-1 & r-2 & \dots & 4 & 3 & 2\\ r & 1 & 2 & 3 & \dots & r-3 & r-2 & r-1 \end{array}\right)=\sigma^{-1} $
The second exercise is
Show that one may take for $\rho$ a permutation that fixes any one of the numbers that $\sigma$ moves (This means: Pick one $1\leq i\leq r$, then one may find a $\rho$ with $\rho(i)=i$)
The only thing I've been able to think of so far is that if you reverse the order of an odd number of numbers, it's always true, e.g.
$\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\ 5 & 4 & 3 & 2 & 1 \end{array} $
The middle column always stays the same. But then there's also even numbers. Ideas?
For even numbers, you'll have to fix a 2nd number, right? For example, if $\sigma=(1 2 3 4)$ and you want to fix $1$, then conjugate by $(2 4)$.