2-generator groups and parabolic class numbers

Newman, Morris

doi:10.1090/S0002-9939-1972-0286896-8

$2$-generator groups and parabolic class numbers
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by Morris Newman

Proc. Amer. Math. Soc. 31 (1972), 51-53

DOI: https://doi.org/10.1090/S0002-9939-1972-0286896-8

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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.

References

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Morris Newman, Classification of normal subgroups of the modular group, Trans. Amer. Math. Soc. 126 (1967), 267–277. MR 204375, DOI 10.1090/S0002-9947-1967-0204375-3

Lehrbuch der Gruppentheorie

11

Bibliographic Information

Journal: Proc. Amer. Math. Soc. 31 (1972), 51-53
DOI: https://doi.org/10.1090/S0002-9939-1972-0286896-8
MathSciNet review: 0286896