[Math] Let $\sigma,\tau \in S_n$. Prove that $\sigma \tau$ and $\tau \sigma $ have the same cycle type.

Question

Let $\sigma,\tau \in S_n$. Prove that $\sigma \tau$ and $\tau \sigma $ have the same cycle type.

I was thinking that you could rewrite $\sigma=g_1\cdots g_k$ with $g_i$ disjoint cycles and $\tau=h_1\cdots h_l$ with $h_i$ disjoint cycles. But I don't know what to do next, as $g_i$ and $h_j$ doesn't have to be disjoint.

Edit: Could I prove it like this ?

We can write $σ$ in disjoint cycles: $σ=σ_1…σ_r$ with lenghts $l_1,…l_r$. So you get: $σ_1=(a_1…a_{l_1}), σ_2=(b_1…b_{l_2}), σ_3=…$.

Which gives:
\begin{align*}
τστ^{-1}&=τσ_1…σ_rτ^{-1}\\
&=τσ_1(τ^{-1}τ)σ_2(τ^{-1}τ)…(τ^{-1}τ)σ_rτ^{-1}\\
&=τ(a_1…a_{l_1})τ^{-1}τ(b_1…b_{l_2})τ^{-1}τ…τσ_rτ^{-1}\\
&=(τ(a_1)…τ(a_{l_1}))(τ(b_1)…τ(b_{l_2}))…τσ_rτ^{-1}\\
\end{align*}

Therefore $σ$ and $τστ^{-1}$ have the same cycle type. So $τσ$ and $στ$ have the same cycle type.

Answer 1

Hint: they are conjugate elements of the group.