Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-15T10:48:38.465Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite metacyclic groups acting on bordered surfaces

Published online by Cambridge University Press:  18 May 2009

Coy L. May
Coy L. May
Affiliation:
Department of Mathematics, Towson State University, Baltimore, Maryland, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A group is called metacyclic in case both its commutator subgroup and commutator quotient group are cyclic. Thus a metacyclic group is a cyclic extension of a cyclic group, and metacyclic groups are among the best understood of the nonabelian groups. Many interesting groups are metacyclic. For instance, the dihedral groups and the “odd” dicyclic groups are metacyclic; see [4, pp. 9–11] for more examples. Here we shall consider the actions of these groups on bordered Klein surfaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 2 , May 1994 , pp. 233 - 240
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Alling, N. L. and Greenleaf, N., Foundations of the theory of Klein surfaces. Lecture Notes in Mathematics Vol. 219, (Springer-Verlag, 1971).CrossRefGoogle Scholar
2.Bujalance, E., Etayo, J. J., Gamboa, J. M., and Gromadzki, G., Automorphism groups of compact bordered Klein surfaces. Lecture Notes in Mathematics Vol. 1439, (Springer-Verlag, 1990).CrossRefGoogle Scholar
3.Bujalance, E. and Martinez, E., A remark on NEC groups of surfaces with boundary. Bull. London Math. Soc. 21 (1989), 263–266.CrossRefGoogle Scholar
4.Coxeter, H. S. M. and Moser, W. O., Generators and Relations for Discrete Groups, Fourth Edition, (Springer-Verlag, 1957).CrossRefGoogle Scholar
5.Gross, J. L. and Lomonaco, S. J., A determination of the toroidal K-metacyclic groups, J. Graph Theory 4 (1980), 165–172.CrossRefGoogle Scholar
6.Hall, M., The theory of groups (Macmillan, 1959).Google Scholar
7.Macbeath, A. M., The classification of non-Euclidean plane crystallographic groups, Canad. J. Math. 19 (1966), 1192–1205.CrossRefGoogle Scholar
8.May, C. L., Automorphisms of compact Klein surfaces with boundary. Pacific J. Math. 59 (1975), 199–210.CrossRefGoogle Scholar
9.May, C. L., Finite groups acting on bordered surfaces and the real genus of a group Rocky Mountain J. Math. 23 (1993), 707–724.CrossRefGoogle Scholar
10.May, C. L., The groups of real genus four, Michigan Math. J. 39 (1992), 219–228.CrossRefGoogle Scholar
11.May, C. L. and Zimmerman, J., The symmetric genus of K-metacyclic groups (to appear).Google Scholar
12.McCullough, D., Minimal genus of abelian actions on Klein surfaces with boundary. Math. Z. 205 (1990), 421–436.CrossRefGoogle Scholar
13.Rotman, J. J., The theory of groups (Allyn and Bacon, 1965).Google Scholar
14.Singerman, D., On the structure of non-Euclidean crystallographic groups, Proc. Cambridge Philos. Soc. 76 (1974), 233–240.CrossRefGoogle Scholar
15.Tucker, T. W., Finite groups acting on surfaces and the genus of a group, J. Combin. Theory Ser. B 34 (1983), 82–98.CrossRefGoogle Scholar
16.Zassenhaus, H., The theory of groups (2nd ed.), (Chelsea, 1958).Google Scholar