Show that $\langle{x,y\,\vert\,x^{2}y^{-2}}\rangle$ is not isomorphic to $S_{3}$

Question

Could someone give me a suggestion to solve this problem?

Show that $\langle{x,y\,\vert\,x^{2}y^{-2}}\rangle$ is not isomorphic to $S_{3}$

Answer 1

$S_3$ is a finite group of order 6, so any group element $g\in S_3$ satisfies $g^6 = 1$.

Can you see why $x^6 \neq 1$ in the group $\langle x, y \;|\; x^2y^{-2}\rangle$?