The symmetric genus of alternating and symmetric groups

Short presentations for alternating and symmetric groups

We derive new families of presentations (by generators and relations) for the alternating and symmetric groups of finite degree n. These include presentations of length that are linear in log n, and 2-generator presentations with a bounded number of relations independent of n.

The Symmetric Genus of the Mathieu Groups

The (symmetric) genus of a finite group may be defined as the smallest genus of those closed orientable surfaces on which G acts faithfully as a group of automorphisms. In this paper the genus of each of the five Mathieu groups Mn, M12, M22, M23 and M24 is determined, with the help of some computer calculations and a little-known theorem of Ree on permutations.

Regular polytopes with symmetric and alternating groups

On the Genus of Symmetric Groups

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 sym...

finite groups which are the products of symmetric or alternating groups with $l_3(4)$

in this paper‎, ‎we determine the simple groups $g=ab$‎, ‎where $b$ is isomorphic to $l_{3}(4)$ and $a$ isomorphic to an alternating or a symmetric group on $ngeq5$‎, ‎letters‎.