The real genus of the alternating groups

  • José Javier Etayo Gordejuela

    Universidad Complutense de Madrid, Spain
  • Ernesto Martínez

    UNED, Madrid, Spain


A Klein surface with boundary of algebraic genus p2\mathfrak{p}\geq 2, has at most 12(p1)12(\mathfrak{p}-1) automorphisms. The groups attaining this upper bound are called MM^{\ast}-groups, and the corresponding surfaces are said to have maximal symmetry. The MM^{\ast}-groups are characterized by a partial presentation by generators and relators. The alternating groups AnA_{n} were proved to be MM^{\ast}-groups when n168n\geq 168 by M. Conder. In this work we prove that AnA_{n} is an MM^{\ast }-group if and only if n13n\geq 13 or n=5,10n=5,10. In addition, we describe topologically the surfaces with maximal symmetry having AnA_{n} as automorphism group, in terms of the partial presentation of the group. As an application we determine explicitly all such surfaces for n14n\leq 14. Each finite group GG 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)\rho(G). If GG is an MM^{\ast}-group then ρ(G)=o(G)12+1\rho(G)=\frac{o(G)}{12}+1. We end our work by calculating the real genus of the alternating groups which are not MM^{\ast}-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