The strong symmetric genus spectrum of abelian groups
type: Item
instance of: scholarly article
title: The strong symmetric genus spectrum of abelian groups
zbMATH Open document ID: 1367.57011
author: Coy L. May
author: Jay Zimmerman
author: Angel V. Kumchev
published in: Archiv der Mathematik
publication date: 2017-05-11T00:00:00Z
review text: The \textit{strong symmetric genus} \(\sigma^0(G)\) of a finite group \(G\) is the smallest genus of the compact Riemann surfaces on which \(G\) has a faithful action as a group of orientation-preserving automorphisms. It is known that every integer \(g \geq 0\) occurs as the strong symmetric genus of some \(G\), see [\textit{C. L. May} and \textit{J. Zimmerman}, Bull. Lond. Math. Soc. 35, No. 4, 433--439 (2003; Zbl 1028.57016)]. In this paper the authors completely determine the set \({\mathcal S}\) of all integers \(g \geq 0\) that occur as \(\sigma^0(G)\) for some abelian group \(G\). Indeed they show that \(g \in {\mathcal S}\) if and only if (a) \(g \equiv 1\) mod 4, or \(g \equiv 55\) mod 81, or (b) \(g-1\) is divisible by \(p^4\) for some odd prime \(p\), or (c) \(g-1\) is divisible by \(a^2\) for some odd integer \(a\) such that \(a-1\) divides \(g\), or (d) \(g-1\) is divisible by \(a^2(a-1)b^2\) for some odd integers \(a, b > 1\) with \(a \equiv 3\) mod 4. This has two interesting corollaries, namely that \({\mathcal S}\) contains no integer \(g \geq 2\) for which \(g-1\) is square-free, and that the range of values of \(\sigma^0\) on abelian groups of rank 5 or more is a subset of the range of \(\sigma^0\) on abelian groups of rank 3 or 4. The authors also prove that the set \({\mathcal S}\) has positive asymptotic density (in the set of non-negative integers), of approximately \(0.3284\).
reviewed by: Marston D. E. Conder
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 6714145
zbMATH Keywords: Riemann surface
zbMATH Keywords: automorphism group
zbMATH Keywords: strong symmetric genus
zbMATH Keywords: genus spectrum
MaRDI profile type: Publication
full work available at URL: s00013-016-1018-8
OpenAlex ID: W2359107735
cites work: Q4691565
cites work: Q3757929
cites work: Q5515488
cites work: Abelian Groups of Automorphisms of Compact Riemann Surfaces
cites work: THERE IS A GROUP OF EVERY STRONG SYMMETRIC GENUS
cites work: Q3423329
cites work: Über die kleinste quadratfreie Zahl einer arithmetischen Reihe
DOI: 10.1007/S00013-016-1018-8
Recommended article: The density of the strong symmetric genus values of p-groups
Recommended article: The Symmetric Genus of p-Groups
Recommended article: Q4691565
Recommended article: The symmetric genus spectrum of abelian groups
Recommended article: The symmetric genus of finite abelian groups
Recommended article: THE STRONG SYMMETRIC GENUS OF DIRECT PRODUCTS
Recommended article: THERE IS A GROUP OF EVERY STRONG SYMMETRIC GENUS
Recommended article: The symmetric genus spectrum of finite groups
Recommended article: Classification of Groups with Strong Symmetric Genus up to Twenty-Five
Recommended article: THE 2-GROUPS OF ODD STRONG SYMMETRIC GENUS
description: scientific article; zbMATH DE number 6714145