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The real genus of the alternating groups

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

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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
Uncontrolled Keywords:alternating groups; real genus; $M^{\ast}$-groups; bordered Klein surfaces
Subjects:Sciences > Mathematics > Group Theory
ID Code:15813
References:

Bujalance, E., Etayo, J. J., Gamboa, J. M. and Gromadzki, G.: Automorphism groups of compact bordered Klein surfaces. A combinatorial approach. Lecture Notes in Mathematics 1439. Springer-Verlag, Berlin, 1990.

Conder, M. D. E.: Minimal generating pairs for permutation groups. Ph.D. Thesis, University of Oxford, 1980.

Conder, M. D. E.: Generators for alternating and symmetric groups. J. London Math. Soc. (2) 22 (1980), 75-86.

Conder, M. D. E.: More on generators for alternating and symmetric groups. Quart. J. Math. Oxford Ser. (2) 32 (1981), 137-163.

Conder, M. D. E.: The symmetric genus of alternating and symmetric groups. J. Combin. Theory Ser. B 39 (1985), 179-186.

Coxeter, H. M. S.: The abstract groups $G^m,n,p$. Trans. Amer. Math. Soc. 45 (1939), 73-150.

Etayo, J. J.: Klein surfaces with maximal symmetry and their groups of automorphisms. Math. Ann. 268 (1984), no. 4, 533-538.

Etayo, J. J.: Una nota sobre $M^\ast$-grupos simétricos y alternados. In Actas XII Jornadas Luso-Españolas de Matemáticas (Braga 1987) II, 62-67, 1989.

Etayo, J. J. and Martínez, E.: Alternating groups, Hurwitz groups and $H^\ast$-groups. J. Algebra 283 (2005), 327-349.

Etayo, J. J. and Martínez, E.: The real genus of cyclic by dihedral and dihedral by dihedral groups. J. Algebra 296 (2006), 145-156.

Greenleaf, N. and May, C. L.: Bordered Klein surfaces with maximal symmetry. Trans. Amer. Math. Soc. 274 (1982), 265-283.

Gromadzki, G. and Mockiewicz, B.: The groups of real genus 6, 7 and 8. Houston J. Math. 28 (2002), 691-699.

May, C. L.: Large automorphism groups of compact Klein surfaces with boundary. I. Glasgow Math. J. 18 (1997), 1-10.

May, C. L.: The species of bordered Klein surfaces with maximal symmetry of low genus. Pacific J. Math. 111 (1984), 371-394.

May, C. L.: The groups of real genus $4$. Michigan Math. J. 39 (1992), 219-228.

May, C. L.: Finite groups acting on bordered surfaces and the real genus of a group. Rocky Mountain J. Math. 23 (1993), 707-724.

May, C. L.: Groups of small real genus. Houston J. Math. 20 (1994), 393-408.

May, C. L.: Real genus actions of finite simple groups. Rocky Mountain J. Math. 31 (2001), 539-551.

May, C. L.: The real genus of $2$-groups. J. Algebra Appl. 6 (2007), no. 1, 103-118.

Miller, G. A.: On the groups generated by two operators. Bull. Amer. Math. Soc. 7 (1901), 424-426.

Singerman, D.: $\rm PSL(2,q)$ as an image of the extended modular group with applications to group actions on surfaces. Proc. Edinburgh Math. Soc. (2) 30 (1987), 143-151.

Wielandt, H.: Finite permutations groups. Academic Press, New York-London, 1964.

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Last Modified:24 Apr 2013 12:41

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