If $G_{1}$ is $\langle a,b \mid a^2 = b^2 \rangle$ and $G_{2}$ is $\langle p,q \mid pqp^{-1} = q^{-1} \rangle$, find an isomorphism $\phi : G_{1} \rightarrow G_{2}$.
I tried the obvious by letting $a$ go to $p$ and $b$ go to $q$, but that did not work.
This appears to be a group theory question; I assume that $G_1, G_2$ are free groups, with the additional relations specified. Note that $(pq)^2=pqpq=pq(q^{-1}p)=p^2$.