In $\langle a, b\mid a^2, b^3, (ab)^2\rangle$, why does $ba=ab^2$?

Question

The Question:

In $\langle a, b\mid a^2, b^3, (ab)^2\rangle$, why does $ba=ab^2$?

My Attempt:

Clearly the presentation defines the group $\mathcal S_3$ under the isomorphism given by $\theta: a\mapsto (12), b\mapsto (123)$ and so $$\begin{align} \theta(ab^2)&=(12)(123)^2 \\ &=(23)(123) \\ &=(13) \\ &=(123)(12) \\ &=\theta(a)\theta(b) \\ &=\theta(ab), \end{align}$$ but what would be a word derivation of $ba$ from $a^2, b^3, (ab)^2$?

Please help :)

Answer 1

\begin{align} abab &= e \\ a^2bab &= a \\ bab &= a \\ bab^3 &= ab^2 \\ ba &= ab^2. \end{align}