# How to show $\langle a, b \; | \; aba = bab \rangle \cong \langle x,y \; | \; x^3=y^2 \rangle$? [duplicate]

Possible Duplicate:
Why do these two presentations present the same group?

I wanted to work my way through the lecture notes of a lecture called "Reflection Groups" and already have problems solving one of the first (probably very simple) exercises given.

Show: $\langle a, b \; | \; aba = bab \rangle \cong \langle x,y \; | \; x^3=y^2 \rangle$.

I already solved a few other, similar exercises, however I just have no idea how to do this one, and before I continue my reading, I wanted to find an answer for that exercise. Wikipedia told me that the latter presentation was a group presentation of $\mathrm{PSL}_2(\mathbb{Z})$, but that does not really help me.

I hope that somebody can help me out and thank you very much in advance for an answer.

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## marked as duplicate by Arturo Magidin, Jyrki Lahtonen♦, t.b., Asaf Karagila, Zhen LinMar 31 '12 at 0:02

Hint: $x\mapsto ab$, $y\mapsto aba$
Noting that $(aba)(bab)=(ab)(ab)(ab)$. – Alex Becker Mar 6 '12 at 13:45