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Título:
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The real genus of the alternating groups
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Autores:
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Etayo Gordejuela, J. Javier ;
Martínez García, Ernesto
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Tipo de documento:
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texto impreso
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Editorial:
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Universidad Autónoma Madrid, 2008
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Palabras clave:
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Estado = Publicado
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Materia = Ciencias: Matemáticas: Grupos (Matemáticas)
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Tipo = Artículo
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Resumen:
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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 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 We end our work by calculating the real genus of the alternating groups which are not $M^{\ast}$-groups.
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