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A Lower Bound for the Real Genusof a Finite Group

Published online by Cambridge University Press:  20 November 2018

Coy L. May
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Abstract

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Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we obtain a good general lower bound for the real genus of the group G. We use the standard representation of G as a quotient of a non-euclidean crystallographic group by a bordered surface group. The lower bound is used to determine the real genus of several infinite families of groups; the lower bound is attained for some of these families. Among the groups considered are the dicyclic groups and some abelian groups. We also obtain a formula for the real genus of the direct product of an elementary abelian 2-group and an “even” dicyclic group. In addition, we calculate the real genus of an abstract family of groups that includes some interesting 3-groups. Finally, we determine the real genus of the direct product of an elementary abelian 2-group and a dihedral group.

Keywords

30F5020H1057M60
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1275 - 1286
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Ailing, N. L. and Greenleaf, N., Foundations of the theory of Klein surfaces, Lecture Notes in Math. 219, Springer-Verlag, Berlin, New York, 1971.Google Scholar
2. Bujalance, E., Etayo, J. J., Gamboa, J. M. and Gromadzki, G., Automorphism groups of compact bordered Klein surfaces, Lecture Notes in Math. 1439, Springer-Verlag, Berlin, New York, 1990.Google Scholar
3. Bujalance, E. and Gromadzki, G., On nilpotent groups of automorphisms of Klein sufaces, Proc. Amer. Math. Soc. 108(1990), 749–759.Google Scholar
4. Coxeter, H. S. M., The abstract groups , Trans. Amer. Math. Soc. 45(1939), 73–150.Google Scholar
5. Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups, Fourth Edition, Springer-Verlag, Berlin, 1957.Google Scholar
6. Macbeath, A. M., The classification of non -Euclidean plane cry stallo graphic groups, Canad. J. Math. 19 (1966), 1192–1205.Google Scholar
7. MacLane, S. and Birkoff, G., Algebra, MacMillan, New York, 1979.Google Scholar
8. May, C. L., Finite groups acting on bordered surfaces and the real genus of a group, Rocky Mountain J. Math. 23(1993), 707–724.Google Scholar
9. May, C. L., The groups of real genus four, Michigan Math. J. 39(1992), 219–228.Google Scholar
10. May, C. L., Finite metacyclic groups acting on bordered surfaces, Glasgow Math. J. 36(1994), 233–240.Google Scholar
11. McCullough, D., Minimal genus of abelian actions on Klein surfaces with boundary,Math. Z. 205(1990), 421–436.Google Scholar
12. Pisanski, T. and White, A. T., Nonorientable embeddings of groups, European J. Combin. 9(1988), 445–461.Google Scholar
13. Rotman, J. J., The Theory of Groups, Allyn and Bacon, Boston, 1965.Google Scholar
14. Singerman, D., On the structure of non-Euclidean cry stallo graphic groups, Math. Proc. Cambridge Philos. Soc. 76(1974), 233–240.Google Scholar
15. White, A. T., Graphs, Groups and Surfaces, Revised Edition, North-Holland, Amsterdam, 1984.Google Scholar