# Hurwitz group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A Hurwitz group is a finite group that occurs (up to isomorphism) as the automorphism group of a Riemann surface of genus $g > 1$, and has the maximum possible order $84(g - 1)$ for such a group. Note that it is in general true that for any Riemann surface of genus $g > 1$, the conformal automorphism group is finite of order at most $84(g - 1)$, and Hurwitz groups are groups where equality is attained. The corresponding Riemann surfaces are termed Hurwitz surfaces.

Note that, in particular, the strong symmetric genus of a Hurwitz group is this genus $g$. Working backward, if the group is $G$, then its strong symmetric genus $\sigma^\circ(G) = g = 1 + (|G|/84)$.

By definition, the order of any Hurwitz group must be a multiple of 84.

Hurwitz groups can be obtained as suitably constructed quotients of the (7,3,2)-von Dyck group.

## Examples

### Small examples

Isomorphism type of Riemann surface Genus $g$ Isomorphism type of group Order of group (equals $84(g - 1)$)
Klein quartic curve 3 projective special linear group:PSL(3,2) 168
Macbeath curve 7 projective special linear group:PSL(2,8) 504