It is well-known [5, 19] that every finite group can appear as a group of automorphisms of an algebraic Riemann surface. Hurwitz [9, 10] showed that the order of such a group can never exceed 84 (g – 1) provided that the genus g is ≧2. In fact, he showed that this bound is the best possible since groups of automorphisms of order 84 (g – 1) are obtainable for some surfaces of genus g. The problems considered by Hurwitz and others can be considered as particular cases of a more general question: Given a finite group G, what is the minimum genus of the surface for which it is a group of automorphisms? This question has been completely answered for cyclic groups by Harvey [7]. Wiman's bound 2(2g + 1), the best possible, materializes as a consequence. A further step was taken by Maclachlan who answered this question for non-cyclic Abelian groups.
Cyclic groups of automorphisms of compact nonorientable Klein surfaces without boundary