On the genus of symmetric groups

Author:
Viera Krňanová Proulx

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

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Abstract | References | Similar Articles | Additional Information

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 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.

**[1]**H. S. M. Coxeter and W. O. J. Moser,*Generators and relations for discrete groups*, 3rd ed., Ergebnisse der Math. und ihrer Grenzgebiete, Bd. 14, Springer-Verlag, Berlin, 1972. MR**0349820 (50:2313)****[2]**J. L. Gross,*Voltage graphs*, Discrete Math.**15**(1974), 239-246. MR**0347651 (50:153)****[3]**J. L. Gross and S. J. Lomonaco, Jr.,*A determination of the toroidal**-metacyclic groups*, J. Graph Theory**4**(1980).**[4]**H. Maschke,*The representation of finite groups*, Amer. J. Math.**18**(1896), 156-194. MR**1505708****[5]**V. K. Proulx,*Classification of the toroidal groups*, Ph.D. Thesis, Columbia Univ., 1977. MR**0480167 (58:354)****[6]**-,*Classification of the toroidal groups*, J. Graph Theory**2**(1978), 269-273. MR**0480167 (58:354)****[7]**T. W. Tucker,*The number of groups of given genus*, Trans. Amer. Math. Soc.**258**(1980), 167-179. MR**554326 (81b:05040)****[8]**-,*Some results on a genus of a group*, J. Graph Theory (submitted).**[9]**A. T. White,*On the genus of a group*, Trans. Amer. Math. Soc.**173**(1972), 203-214. MR**0317980 (47:6529)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0617549-1

Keywords:
Cayley graph,
genus of a group,
symmetric group,
voltage graphs,
Euler characteristic

Article copyright:
© Copyright 1981
American Mathematical Society