A von Dyck group is a group with presentation $< a,b | a^m=b^n=(ab)^p=1 >$ with m,n,p natural numbers. Is it known which of these groups are solvable and which are not? Is there a reference for this? Thanks.
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1$\begingroup$ The groups with $\frac{1}{m} + \frac{1}{n} + \frac{1}{p} > 1$ are all finite, and all solvable except $\Delta(2,3,5) \cong A_5$. The groups with $\frac{1}{m} + \frac{1}{n} + \frac{1}{p} = 1$ are all infinite and solvable: their commutator subgroups are isomorphic to $\mathbb{Z}^2$. The groups with $\frac{1}{m} + \frac{1}{n} + \frac{1}{p} < 1$ are all infinite and nonsolvable: indeed, with only finitely many exceptions, each of these groups has a simple group $PSL_2(\mathbb{F}_q)$ as a quotient. $\endgroup$– Pete L. ClarkCommented Jun 21, 2010 at 2:27
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$\begingroup$ Thank you for the comment. Do you know where I can find a proof for the case of 1/m + 1/n + 1/p = 1? $\endgroup$– daveCommented Jun 21, 2010 at 3:41
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$\begingroup$ These days, these groups are usually called triangle groups. You might like to look at the answers to the following question, which was very similar. mathoverflow.net/questions/22459/x-y-xp-yp-xyp-1/22463#22463 $\endgroup$– HJRWCommented Jun 22, 2010 at 21:27
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1 Answer
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You might try Generators and Relations for Discrete Groups by Coxeter and Moser.
Specifically for 1/m + 1/n + 1/p = 1 there are only 3 cases up to permutation, (2,3,6), (2,4,4) and (3,3,3). Map a and b to an appropriate root of unity to get a homomorphism onto C_6, C_4, or C_3, respectively. The kernel of the map is in all three cases isomorphic to Z^2.
