Hurwitz's Automorphisms Theorem - Examples of Hurwitz's Groups and Surfaces

Examples of Hurwitz's Groups and Surfaces

The smallest Hurwitz group is the projective special linear group PSL(2,7), of order 168, and the corresponding curve is the Klein quartic curve. This group is also isomorphic to PSL(3,2).

Next is the Macbeath curve, with automorphism group PSL(2,8) of order 504. Many more finite simple groups are Hurwitz groups; for instance all but 64 of the alternating groups are Hurwitz groups, the largest non-Hurwitz example being of degree 167. The smallest alternating group that is a Hurwitz group is A₁₅.

Most projective special linear groups of large rank are Hurwitz groups, (Lucchini, Tamburini & Wilson 2000). For lower ranks, fewer such groups are Hurwitz. For n_p the order of p modulo 7, one has that PSL(2,q) is Hurwitz if and only if either q=7 or q = pn_p. Indeed, PSL(3,q) is Hurwitz if and only if q = 2, PSL(4,q) is never Hurwitz, and PSL(5,q) is Hurwitz if and only if q = 74 or q = pn_p, (Tamburini & Vsemirnov 2006).

Similarly, many groups of Lie type are Hurwitz. The finite classical groups of large rank are Hurwitz, (Lucchini & Tamburini 1999). The exceptional Lie groups of type G2 and the Ree groups of type 2G2 are nearly always Hurwitz, (Malle 1990). Other families of exceptional and twisted Lie groups of low rank are shown to be Hurwitz in (Malle 1995).

There are 12 sporadic groups that can be generated as Hurwitz groups: the Janko groups J₁, J₂ and J₄, the Fischer groups Fi₂₂ and Fi'₂₄, the Rudvalis group, the Held group, the Thompson group, the Harada–Norton group,the third Conway group Co₃, the Lyons group, and the Monster, (Wilson 2001).