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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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: 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.


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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

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