The real genus of the alternating groups.
type: Item
instance of: scholarly article
title: The real genus of the alternating groups.
zbMATH Open document ID: 1173.20024
DOI: 10.4171/RMI/558
author: José Javier Etayo
author: Ernesto Martínez
published in: Revista Matemática Iberoamericana
publication date: 2009-02-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1228834296
full work available at URL: https://eudml.org/doc/42398
review text: A Klein surface with boundary of algebraic genus \(\mathfrak p\geq 2\), has at most \(12(\mathfrak p-1)\) automorphisms. The groups attaining this upper bound are called \(M^*\)-groups, and the corresponding surfaces are said to have maximal symmetry. The \(M^*\)-groups are characterized by a partial presentation by generators and relators. The alternating groups \(A_n\) were proved to be \(M^*\)-groups when \(n\geq 168\) by M. Conder. In this work the authors prove that \(A_n\) is an \(M^*\)-group if and only if \(n\geq 13\) or \(n=5\), \(10\). In addition, they describe topologically the surfaces with maximal symmetry having \(A_n\) as automorphism group, in terms of the partial presentation of the group. As an application they determine explicitly all such surfaces for \(n\leq 14\). Each finite group \(G\) acts as an automorphism group of several Klein surfaces. The minimal genus of these surfaces is called the real genus of the group, \(\rho(G)\). If \(G\) is an \(M^*\)-group then \(\rho(G)=\tfrac{o(G)}{12}+1\).
reviewed by: Osman Bizim
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 5509265
zbMATH Keywords: alternating groups
zbMATH Keywords: real genera
zbMATH Keywords: \(M^*\)-groups
zbMATH Keywords: bordered Klein surfaces
zbMATH Keywords: partial presentations
zbMATH Keywords: generators
zbMATH Keywords: relators
zbMATH Keywords: automorphism groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5509265