THE SYMMETRIC GENUS OF LARGE ODD ORDER GROUPS

let p be an odd prime, and let L be the set of integers n for which there is a solvable group G of order n generated by elements of order p. We prove that the set L has density 0 in the set of positive integers. This result generalizes work on the symmetric genus of an odd order group. Suppose the odd order group G acts on a Riemann surface of genus g >= 2. If vertical bar G vertical bar > 8(g - 1), then vertical bar G vertical bar = K (g - 1), where K is 15, 21/2, 9 or 33/4. We call these four types of groups LO-1 groups through LO-4 groups, respectively. These groups are quotients of Fuchsian triangle groups of type (3, 3, n), for n = 5, 7, 9 and 11, respectively. Since each LO-group is generated by two elements of order 3 and the genus of an LO-group is determined by its order, the set of integers g for which there is a LO - group of symmetric genus g has density 0 in the set of positive integers. We also obtain restrictions on the powers of the primes dividing the orders of LO groups. In addition, we study the metabelian LO-3 groups and obtain information about their group theoretic structure. This allows us to classify the integers that are the orders of metabelian LO-3 groups and to identify all such groups.