The real genus of direct products Zn x G

• Autores: Coy L. May
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 35, Nº 1, 2009, págs. 23-37
• Idioma: inglés
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• Resumen

  • Let G be a finite group. The real genus of G is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we consider the problem of finding the real genus of the direct product Zn× G , in which one factor is cyclic and the other factor G is a familiar group with well-known properties or, alternately, a group for which the real genus has already been determined. We focus on groups that are generated by two elements, one of which is an involution. Let G be a finite non-abelian group of this type, and assume the order n of the cyclic factor is relatively prime to |G|. Our main result is the determination of the real genus of this direct product in terms of n, |G|, and two parameters associated with the group G. We then give a range of applications of this result. Each genus formula yields a sequence of values for the real genus g such that there exists a group of genus g.