# Revista Matemática Iberoamericana

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**Volume 24, Issue 3, 2008, pp. 865–894**

**DOI: 10.4171/RMI/558**

Published online: 2008-12-31

The real genus of the alternating groups

José Javier Etayo Gordejuela^{[1]}and Ernesto Martínez

^{[2]}(1) Universidad Complutense de Madrid, Spain

(2) UNED, Madrid, Spain

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.

*Keywords: *Alternating groups, real genus, *M**-groups, bordered Klein surfaces

Etayo Gordejuela José Javier, Martínez Ernesto: The real genus of the alternating groups. *Rev. Mat. Iberoamericana* 24 (2008), 865-894. doi: 10.4171/RMI/558