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Groups Complexity Cryptology

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Editorial Board: Ciobanu, Laura / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

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An update on Hurwitz groups

Marston Conder
  • Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand. E-mail:
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Published Online: 2010-06-23 | DOI: https://doi.org/10.1515/gcc.2010.002


A Hurwitz group is any non-trivial finite quotient of the (2, 3, 7) triangle group, that is, any non-trivial finite group generated by elements x and y satisfying x 2 = y 3 = (xy)7 = 1. Every such group G is the conformal automorphism group of some compact Riemann surface of genus g > 1, with the property that |G| = 84(g – 1), which is the maximum possible order for given genus g. This paper provides an update on what is known about Hurwitz groups and related matters, following up the author's brief survey in Bull. Amer. Math. Soc.23 (1990).

Keywords.: Hurwitz groups; triangle group; (2,3,7); Riemann surfaces

About the article

Received: 2008-04-25

Published Online: 2010-06-23

Published in Print: 2010-06-01

Citation Information: Groups – Complexity – Cryptology, Volume 2, Issue 1, Pages 35–49, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc.2010.002.

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