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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 Lin Mar 31 '12 at 0:02

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Are these notes online? If so could you post the link? I would be interested in taking a look –  Edison Mar 6 '12 at 13:48
    
@Edison: The notes are online, but in German. (math.ethz.ch/~suter/refl.pdf) –  Huy Mar 6 '12 at 13:52

1 Answer 1

up vote 6 down vote accepted

Hint: $x\mapsto ab$, $y\mapsto aba$

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How does one come up with such a map? –  Huy Mar 6 '12 at 13:44
3  
Noting that $(aba)(bab)=(ab)(ab)(ab)$. –  Alex Becker Mar 6 '12 at 13:45
    
@Huy: What Alex says. Experience with reflection groups also helps :-) –  Jyrki Lahtonen Mar 6 '12 at 13:48

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