Suppose $\displaystyle a$ and $\displaystyle b$ belong to a group, $\displaystyle a$ has odd order, and $\displaystyle aba^{-1} = b^{-1}$. Show that $\displaystyle b^2 = e$.

I have a feeling I'm missing something obvious, but I can't for the life of me figure out what $\displaystyle a$ having odd order has to do with this.