Etayo Gordejuela, J. Javier and Martínez García, Ernesto
(2008)
*The real genus of the alternating groups.*
Revista Matemática Iberoamericana, 24
(3).
pp. 865-894.
ISSN 0213-2230

Official URL: http://projecteuclid.org/euclid.rmi/1228834296

## Abstract

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.

Item Type: | Article |
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Uncontrolled Keywords: | alternating groups; real genus; $M^{\ast}$-groups; bordered Klein surfaces |

Subjects: | Sciences > Mathematics > Group Theory |

ID Code: | 15813 |

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Deposited On: | 03 Jul 2012 09:14 |

Last Modified: | 24 Apr 2013 12:41 |

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