On the genus of symmetric groups
HTML articles powered by AMS MathViewer
- by Viera Krňanová Proulx PDF
- Trans. Amer. Math. Soc. 266 (1981), 531-538 Request permission
Abstract:
A new method for determining genus of a group is described. It involves first getting a bound on the sizes of the generating set for which the corresponding Cayley graph could have smaller genus. The allowable generating sets are then examined by methods of computing average face sizes and by voltage graph techniques to find the best embeddings. This method is used to show that genus of the symmetric group ${S_5}$ is equal to four. The voltage graph method is used to exhibit two new embeddings for symmetric groups on even number of elements. These embeddings give us a better upper bound than that previously given by A. T. White.References
- H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14, Springer-Verlag, New York-Heidelberg, 1972. MR 0349820
- Jonathan L. Gross, Voltage graphs, Discrete Math. 9 (1974), 239–246. MR 347651, DOI 10.1016/0012-365X(74)90006-5 J. L. Gross and S. J. Lomonaco, Jr., A determination of the toroidal $K$-metacyclic groups, J. Graph Theory 4 (1980).
- H. Maschke, The Representation of Finite Groups, Especially of the Rotation Groups of the Regular Bodies of Three-and Four-Dimensional Space, by Cayley’s Color Diagrams, Amer. J. Math. 18 (1896), no. 2, 156–194. MR 1505708, DOI 10.2307/2369680
- Viera Krňanová Proulx, Classification of the toroidal groups, J. Graph Theory 2 (1978), no. 3, 269–273. MR 480167, DOI 10.1002/jgt.3190020312
- Viera Krňanová Proulx, Classification of the toroidal groups, J. Graph Theory 2 (1978), no. 3, 269–273. MR 480167, DOI 10.1002/jgt.3190020312
- T. W. Tucker, The number of groups of a given genus, Trans. Amer. Math. Soc. 258 (1980), no. 1, 167–179. MR 554326, DOI 10.1090/S0002-9947-1980-0554326-3 —, Some results on a genus of a group, J. Graph Theory (submitted).
- Arthur T. White, On the genus of a group, Trans. Amer. Math. Soc. 173 (1972), 203–214. MR 317980, DOI 10.1090/S0002-9947-1972-0317980-2
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 266 (1981), 531-538
- MSC: Primary 05C10; Secondary 05C25, 20B05, 20F32
- DOI: https://doi.org/10.1090/S0002-9947-1981-0617549-1
- MathSciNet review: 617549