# The real genus of the alternating groups

### José Javier Etayo Gordejuela

Universidad Complutense de Madrid, Spain### Ernesto Martínez

UNED, Madrid, Spain

## Abstract

A Klein surface with boundary of algebraic genus $p≥2$, has at most $12(p−1)$ 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 $n≥168$ by M. Conder. In this work we prove that $A_{n}$ is an $M_{∗}$-group if and only if $n≥13$ 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 $n≤14$. 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, $ρ(G)$. If $G$ is an $M_{∗}$-group then $ρ(G)=12o(G) +1$. We end our work by calculating the real genus of the alternating groups which are not $M_{∗}$-groups.

## Cite this article

José Javier Etayo Gordejuela, Ernesto Martínez, The real genus of the alternating groups. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 865–894

DOI 10.4171/RMI/558