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The symmetric cross-cap number of the groups Cm × Dn

Published online by Cambridge University Press:  12 November 2008

J. J. Etayo Gordejuela  and
E. Martínez
J. J. Etayo Gordejuela
Affiliation:
Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain ([email protected])
E. Martínez
Affiliation:
Departamento de Mathemáticas, Universidad Nacional de Educación a Distancia, Paseo Senda del Rey 9, 28040 Madrid, Spain ([email protected])
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Abstract

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric cross-cap number and denoted by . This number is related to other parameters defined on surfaces as the symmetric genus and the strong symmetric genus.

The systematic study of the symmetric cross-cap number was begun by C. L. May, who also calculated it for certain finite groups. Here we obtain the symmetric cross-cap number for the groups Cm × Dn. As an application of this result, we obtain arithmetic sequences of integers which are the symmetric cross-cap number of some group. Finally, we recall the several different genera of the groups Cm × Dn.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 138 , Issue 6 , December 2008 , pp. 1197 - 1213
Copyright
Copyright © Royal Society of Edinburgh 2008

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