The symmetric genus of finite abelian groups
type: Item
instance of: scholarly article
title: The symmetric genus of finite abelian groups
zbMATH Open document ID: 0805.30035
author: Coy L. May
author: Jay Zimmerman
published in: Illinois Journal of Mathematics
publication date: 1995-02-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 20K99
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 561220
zbMATH Keywords: symmetric genus
reviewed by: C. Maclachlan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 561220