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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

$ 2$-generator groups and parabolic class numbers


Author: Morris Newman
Journal: Proc. Amer. Math. Soc. 31 (1972), 51-53
DOI: https://doi.org/10.1090/S0002-9939-1972-0286896-8
MathSciNet review: 0286896
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Abstract: It is shown that if $ x,y$ are generators of the finite group $ G$ such that $ {x^p} = {y^q} = {(xy)^n} = 1$, where $ p,q,n$ are integers $ > 1,(p,q) = 1$, and $ xy$ is of true order $ n$, then the order $ \mu = nt$ of $ G$ satisfies $ n \leqq pq{t^p}$. This result is used to show that if $ F$ is a Fuchsian group of genus 0 generated by 2 elliptic elements of coprime order and with 1 parabolic class, then $ F$ possesses only finitely many normal subgroups having a given number of parabolic classes.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0286896-8
Keywords: Parabolic classes, Fuchsian groups, $ 2$-generator groups
Article copyright: © Copyright 1972 American Mathematical Society