Hmm...

Suppose that $\displaystyle xy^{4}=y^{4}x$ is true and that $\displaystyle xy^{2} = y^{3}x$ and $\displaystyle yx^{3} = x^{2}y$. Then

$\displaystyle xy^{4} = y^{6}x = y^{4}x \Rightarrow y^{2}= e \Rightarrow x = yx \Rightarrow y \,\,\text{is the identity}$

But then $\displaystyle x^{3} = x^{2} \Rightarrow x \,\,\text{is the identity.}$ This would be a consequence of these relations; does this make sense in the group your considering? I only ask because its nice to know that it's plausible to derive the first relation from the other two before you begin. If $\displaystyle x,y$ are definitely not the identity element then you needn't waste your time!

Please make sure I haven't made a silly mistake because I excel at those.