Published online by Cambridge University Press: 18 May 2009
In the nineteenth century, Hurwitz  and Wiman  obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, , . These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey  and Maclachlan , respectively, in the case of Riemann surfaces and by Bujalance , Hall  and Gromadzki  in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf )—was minimized by Bujalance- Etayo-Gamboa-Martens  in the case where G is cyclic and by McCullough  in the abelian case.