Conjugation of permutations proof

Question

I'm trying to prove the following statement:

Let $f,g \in S_n$. Let $g = c_1c_2\dots c_k$ be the disjunct cyclus decomposition of $g$. Then the disjunct cyclus decomposition of $fgf^{-1}$ is found by replacing $a_r$ by $f(a_r)$ in every cyclus $c_i = (a_1,..,a_l)$.

Important remark: I use the convention that we apply permutations from right to left.

My attempt:

I don't know good what I should do: I tried the following

$fgf^{-1} = fc_1c_2\dots c_kf^{-1} = (fc_1f^{-1})(fc_2f^{-1})\dots(fc_kf^{-1})$

and then I want to show that $(fc_if^{-1}) = (f(a_1), \dots ,f(a_l))$ but I'm not sure how to proceed or how to prove this. Can someone give a hint or proof?

Thanks in advance.

Answer 1

If we have a cycle $(a_1\ a_2\ \cdots\ a_l)$ of $g$, set $b_i=f(a_i)$. Then $$(fgf^{-1})(b_i)=fgf^{-1}f(a_i)=fg(a_i)=f(a_{i+1})=b_{i+1}$$ except, of course, when $i=l$, and we get $b_1$ instead. So $(b_1\ b_2\ \cdots\ b_l)$ is a cycle of $fgf^{-1}$.