Proving Injectivity of Group Homomorphism Given Relations

[a.powell]

Homework Statement

Let $G = \langle x,y \ | \ x^2, y^3, (xy)^3 \rangle$, and $f: G \rightarrow A_4$ the unique homomorphism such that $f(x) = a$, $f(y) = b$, where $a = (12)(34)$ and $b = (123)$. Prove that $f$ is an isomorphism. You may assume that it is surjective.

Homework Equations

N/A

The Attempt at a Solution

The previous stages of the question proved that the stated homomorphism was in fact unique, and considered the subgroup $H = \langle y \rangle$ along with the set of cosets $A = \{H, xH, yxH, y^2xH\}$, from which I have proven that $gH \in A, \ \forall g \in G$. I'm not sure what relevance those stages have to the proof of isomorphism though I'd assume there is some relevance since otherwise the question would be a little disjoint; my only thought so far is that it relates to the first isomorphism theorem for groups, since we have a group homomorphism and a set of cosets, but I can't see how to make it work in this example. We did a similar question in lectures involving proving that $D_{2n} \cong \langle x,y \ | \ x^n, y^2, (xy)^2 \rangle$ where injectivity was proven by rewriting elements of the group as words of the form $x^ky^l$ using the given relations, and concluding that $\#G \leq 2n$. I have tried something similar with this question but am struggling to rewrite elements of $G$ in a similar form (though I'd guess there'd be an irreducible factor of $(xy)^m$ in the word somewhere) which is useful.

 

[a.powell]

I hate to bump, but...anyone?