The Hurwitz bound for a compact Riemann surface 
of genus 
is the upper bound
on the group order of its automorphism group, as established by Hurwitz's
automorphism theorem (Hurwitz 1893). The bound is sharp. A compact Riemann surface attaining equality is called a Hurwitz
surface; examples include the Klein quartic of genus 3 and the Fricke-Macbeath
curve of genus 7 (Hurwitz 1893, Macbeath 1965).
See also
Fricke-Macbeath Curve,
Hurwitz Map,
Hurwitz's
Automorphism Theorem,
Klein Quartic,
Riemann
Surface
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References
Hurwitz, A. "Über algebraische Gebilde mit eindeutigen Transformationen in sich." Math. Ann. 41, 403-442, 1893. https://doi.org/10.1007/BF01443420.Macbeath,
A. M. "On a Curve of Genus 7." Proc. London Math. Soc. 15,
527-542, 1965. https://doi.org/10.1112/plms/s3-15.1.527.
Cite this as:
Weisstein, Eric W. "Hurwitz Bound." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HurwitzBound.html
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