Let $G$ be a group in which $aba = b \; \forall a,b \in G$. Prove that $G$ is Abelian.

You're correct.

If $a^2=e$ for all $a\in G$, then for any $g,h\in G$,

$$\begin{align} (gh)^2&=ghgh\\ &=e\\ &=ee\\ &=g^2h^2\\ &=gghh, \end{align}$$

from which it follows that $gh=hg$. Hence $G$ is abelian.