im trying to prove the following On the set $S_n$ of permutations define $σ_1 \sim σ_2$ if there exists a permutation τ such that $σ_1 = τσ_2τ^{-1}$
Show that $\sim$ defines an equivalence relation on the set $S_n$of permutations.
so this might be a dumb question but im confused if i can left/right multiply like i would with matrices for example can do the following:
$σ_1 \sim σ_2$ implies $σ_1 = τσ_2τ^{-1} \implies$ $τ^{-1}σ_1 = τ^{-1}τσ_2τ^{-1} \implies τ^{-1}σ_1 = idσ_2τ^{-1}$
this is the step where i am not sure if i can do the following
$τ^{-1}σ_1τ = σ_2τ^{-1}τ \implies τ^{-1}σ_1τ = σ_2 id $
and this shows that
$σ_2 \sim σ_1$
am i allowed to do this in order to show the relation?
Yes, you can multiply an equation either from right or from left with the same element, and you'll still get an equation.
$$\sigma_1=\tau\sigma_2\tau^{-1}\ \iff\ \tau^{-1}\sigma_1\tau=\sigma_2$$
So, $\tau^{-1}$ will confirm $\sigma_2\sim\sigma_1$.