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The following question is my homework and I cannot solve it myself. Could somebody please help me with this one? Thank you so much

Prove that the group $G=\langle x,y\mid x^m,y^n\rangle$ is infinite when $m,n\geq2$.

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Hi Taukhtn. When you ask a question, please do not pose it with a demanding tone. Do not say "Prove this!" or "Prove that!" In order to conform to MSE etiquette, try to describe why your question is giving you trouble and how you would like to seek help from others. As you are new to this community, I am giving you advice on how to do things the courteous way, so that your experience with the MSE will be a very satisfying one. :) Cool? – Haskell Curry Dec 12 '12 at 8:11
I really appreciate your advice and I am really sorry for any inconvenience may cause. This is my first time joining a forum hence i lack of experience. Again, thank you for your advice – Taukhtn Dec 12 '12 at 9:53
Every group $H$ which is generated by an element of order $m$ and an element of order $n$ is a quotient of this group $G$. How many such groups $H$ are there? – user1729 Dec 12 '12 at 10:04

1 Answer

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HINT:

Consider elements of the form $x, xy, xyx, xyxy, \ldots$

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I have done this way but it didn't work :-s – Taukhtn Dec 12 '12 at 7:19
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@Apollo: In what way did it not work? – Brian M. Scott Dec 12 '12 at 7:20
I tried to prove that $xy$ has infinite order, but i cannot – Taukhtn Dec 12 '12 at 7:31
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It's very hard to help, because we do not know how much you know about the theory of finitely presented groups. If you have come across free products, then the question is easy. – Derek Holt Dec 12 '12 at 9:36
I have learnt about free group and presentation only, is it enough for me to solve this problem? Thank you so much :) – Taukhtn Dec 12 '12 at 11:45
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