Hurwitz group

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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.


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

Families of examples