Geodesic
A Lower Bound for the Real Genusof a Finite Group
Canadian journal of mathematics, Tome 46 (1994) no. 6, pp. 1275-1286

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-1994-072-x
Mots-clés : 30F50, 20H10, 57M60 
A Lower Bound for the Real Genusof a Finite Group. Canadian journal of mathematics, Tome 46 (1994) no. 6, pp. 1275-1286. doi: 10.4153/CJM-1994-072-x
@misc{10_4153_CJM_1994_072_x,
     title = {A {Lower} {Bound} for the {Real} {Genusof} a {Finite} {Group}},
     journal = {Canadian journal of mathematics},
     pages = {1275--1286},
     year = {1994},
     volume = {46},
     number = {6},
     doi = {10.4153/CJM-1994-072-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1994-072-x/}
}
TY  - JOUR
TI  - A Lower Bound for the Real Genusof a Finite Group
JO  - Canadian journal of mathematics
PY  - 1994
SP  - 1275
EP  - 1286
VL  - 46
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1994-072-x/
DO  - 10.4153/CJM-1994-072-x
ID  - 10_4153_CJM_1994_072_x
ER  - 
%0 Journal Article
%T A Lower Bound for the Real Genusof a Finite Group
%J Canadian journal of mathematics
%D 1994
%P 1275-1286
%V 46
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1994-072-x/
%R 10.4153/CJM-1994-072-x
%F 10_4153_CJM_1994_072_x

[1] 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] 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] 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] 4. Coxeter, H. S. M., The abstract groups , Trans. Amer. Math. Soc. 45(1939), 73–150. Google Scholar

[5] 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] 6. Macbeath, A. M., The classification of non -Euclidean plane cry stallo graphic groups, Canad. J. Math. 19 (1966), 1192–1205. Google Scholar

[7] 7. MacLane, S. and Birkoff, G., Algebra, MacMillan, New York, 1979. Google Scholar

[8] 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] 9. May, C. L., The groups of real genus four, Michigan Math. J. 39(1992), 219–228. Google Scholar

[10] 10. May, C. L., Finite metacyclic groups acting on bordered surfaces, Glasgow Math. J. 36(1994), 233–240. Google Scholar

[11] 11. McCullough, D., Minimal genus of abelian actions on Klein surfaces with boundary,Math. Z. 205(1990), 421–436. Google Scholar

[12] 12. Pisanski, T. and White, A. T., Nonorientable embeddings of groups, European J. Combin. 9(1988), 445–461. Google Scholar

[13] 13. Rotman, J. J., The Theory of Groups, Allyn and Bacon, Boston, 1965. Google Scholar

[14] 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] 15. White, A. T., Graphs, Groups and Surfaces, Revised Edition, North-Holland, Amsterdam, 1984. Google Scholar

Cité par Sources :