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oai:docta.ucm.es:20.500.14352/500342024-07-11T13:26:22Zcom_20.500.14352_14col_20.500.14352_15

The real genus of the alternating groups Etayo Gordejuela, José Javier Martínez García, Ernesto 512.54 Alternating groups Real genus $M^{\ast}$-groups Bordered Klein surfaces Grupos (Matemáticas) 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. Depto. de Álgebra, Geometría y Topología Fac. de Ciencias Matemáticas TRUE pub 2023-06-20T09:36:50Z 2023-06-20T09:36:50Z 2008 journal article https://hdl.handle.net/20.500.14352/50034 0213-2230 metadata only access Universidad Autónoma Madrid