Hurwitz group
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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.