Metacyclic groups as automorphism groups of compact Riemann surfaces

Abstract

Let X be a compact Riemann surface of genus , and let G be a subgroup of . We show that if the Sylow 2-subgroups of G are cyclic, then . If all Sylow subgroups of G are cyclic, then, with two exceptions, . More generally, if G is metacyclic, then, with one exception, . Each of these bounds is attained for infinitely many values of g.