# 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 propertiesVIEW 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 , *and* has the maximum possible order for such a group. Note that it is in general true that for any Riemann surface of genus , the conformal automorphism group is finite of order at most , 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 . Working backward, if the group is , then its strong symmetric genus .

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 | Isomorphism type of group | Order of group (equals ) |
---|---|---|---|

Klein quartic curve | 3 | projective special linear group:PSL(3,2) | 168 |

Macbeath curve | 7 | projective special linear group:PSL(2,8) | 504 |

### Families of examples

- Alternating groups of sufficiently large degree are Hurwitz groups: All except 64 alternating groups are Hurwitz groups. The smallest Hurwitz alternating group is alternating group:A15 and the largest
*non-Hurwitz*alternating group is the alternating group of degree 167. - There are congruence conditions that describe when the projective special linear group of degree two over a finite field is Hurwitz.
- Twelve of the sporadic simple groups are Hurwitz: Janko group:J1, Janko group:J2, Janko group:J4, Fischer group:Fi22, and others.