The symmetric genus of finite abelian groups
The symmetric genus of a finite group \(G\) is the minimum genus of any compact Riemann surface on which \(G\) acts faithfully. In this paper, the symmetric genus of any finite abelian group \(A\) is determined. If one restricts to orientation -- preserving actions -- the strong symmetric genus -- this was determined by the reviewer [Proc. Lond. Math. Soc. 15, 699-712 (1965; Zbl 0156.089)]. See also \textit{K. Hayakawa} [Bull. Fac. Sci. Eng. Ohio Univ. Math. 36, 15-20 (1993; reviewed above)]. Clearly, if the order of \(A\) is odd, then the two coincide. The formulas obtained here for the symmetric genus show that it depends on the ranks of \(A\) and of the elementary abelian 2-part of \(A\) in its canonical form. The method is to express \(A\) as a quotient of an NEC group and to use structure theorem for NEC groups.
scientific article; zbMATH DE number 561220