On the monodromy groups of Riemann surfaces of genus zero

Abstract

It is known that every finite group G can appear as the monodromy group of some Riemann surface of genus ⩾0. The fact that symmetric groups of all orders can appear as monodromy groups of Riemann surfaces of genus zero is a long-standing one. In this paper, a further search has been made in order to determine which finite groups G can and cannot appear as monodromy groups of Riemann surfaces of genus zero. It has been shown, on the one hand, that every alternating group, the simple group PSL(2, 7) and all cyclic and dihedral groups can appear as such monodromy groups by using a right coset representation of each with respect to a particular subgroup. It has been shown, on the other hand, that the quaternion group, the generalized quaternion group of order 16, a non-Abelian group of order 27 with specified presentation as well as every finite direct product X i = 1 n Z m i with m i | m i + 1 and n > 1 of cyclic groups of order >4 can never appear as such monodromy groups using right coset representations with respect to subgroups. The examples cited above suggest the conjecture that all simple groups can appear as monodromy groups (right coset representations with respect to subgroups being employed to determine the monodromy group) of Riemann surfaces of genus zero and that noncyclic, solvable groups cannot appear unless they are extensions of degree 2 of a cyclic group.