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November, 2008 The real genus of the alternating groups
José Javier Etayo Gordejuela , Ernesto Martínez
Rev. Mat. Iberoamericana 24(3): 865-894 (November, 2008).

Abstract

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 $M^{\ast}$-groups, and the corresponding surfaces are said to have maximal symmetry. The $M^{\ast}$-groups are characterized by a partial presentation by generators and relators. The alternating groups $A_{n}$ were proved to be $M^{\ast}$-groups when $n\geq 168$ by M. Conder. In this work we prove that $A_{n}$ is an $M^{\ast }$-group if and only if $n\geq 13$ or $n=5,10$. In addition, we 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 we 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 $M^{\ast}$-group then $\rho(G)=\frac{o(G)}{12}+1$. We end our work by calculating the real genus of the alternating groups which are not $M^{\ast}$-groups.

Citation

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José Javier Etayo Gordejuela . Ernesto Martínez . "The real genus of the alternating groups." Rev. Mat. Iberoamericana 24 (3) 865 - 894, November, 2008.

Information

Published: November, 2008
First available in Project Euclid: 9 December 2008

zbMATH: 1173.20024
MathSciNet: MR2490164

Subjects:
Primary: 20F05
Secondary: 30F10

Keywords: $M^{\ast}$-groups , alternating groups , bordered Klein surfaces , real genus

Rights: Copyright © 2008 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.24 • No. 3 • November, 2008
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