Show that the class of the word $aba$ in group $\langle{a,b\,\vert\,a^{2}b^{2} }\rangle$ is not the trivial element.

Question

Could anyone give me a suggestion to solve this problem about group presentations?

Show that the class of the word $aba$ in group $\langle{a,b\mid a^{2}b^{2} }\rangle$ is not the trivial element.

Answer 1

There is a homomorphism from your group to the cyclic group of order 2 which maps the two generators to the nonzero element and aba to a nonzero element.

Answer 2

As a matter of strategy, try to find some abelian quotient in which $2a + b \ne 0$ (using additive notation). Note (in your quotient) $2a + b = -b$. So now it suffices to find an abelian quotient in which $b \ne 0$.