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On non-Hurwitz groups and non-congruence subgroups of the modular group

Published online by Cambridge University Press:  18 May 2009

Jeffrey Cohen
Jeffrey Cohen
Affiliation:
University of Pittsburgh, Pittsburgh, Pennsylvania 15260
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In this note homomorphisms of (2, 3, n) = 〈x, y: x2 = y3 = (xy)n = 1) into PSL3(q) are considered. Of particular interest is (2, 3, 7) whose finite factors are referred to as Hurwitz groups. It is known (see [3]) that for certain q, PSL2(q) is a Hurwitz group, so that one might suppose that PSL3(q) is a natural place to search for new Hurwitz groups. This intuition turns out to be ill-founded, for as we shall see all Hurwitz subgroups of PSL3(q) have already been discovered in [3].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 22 , Issue 1 , January 1981 , pp. 1 - 7
Copyright
Copyright © Glasgow Mathematical Journal Trust 1981

References

REFERENCES

1.Hartley, W. W., Ternary collineation groups, Ann. of Math. 27 (1925), 140–158.CrossRefGoogle Scholar
2.Leech, John, Generators for certain normal subgroups of (2, 3, 7), Proc. Cambridge Philos. Soc. 61 (1965), 321–332.CrossRefGoogle Scholar
3.Macbeath, A. M., Generators of the linear fractional groups, Proceedings of a Symposium of Pure Mathematics in Number Theory, Vol. XII, Houston (1967), 14–32.Google Scholar
4.Mitchell, H. H., Ternary linear groups, Trans. Amer. Math. Soc. 12 (1911), 207–242.CrossRefGoogle Scholar
5.Newman, Morris, Maximal normal subgroups of the modular group, Proc. Amer. Math. Soc. 19 (1968), 1138–44.CrossRefGoogle Scholar