ALTERNATING AND SYMMETRIC GROUPS WITH EULERIAN GENERATING GRAPH

Generating Symmetric and Alternating Groups with Elements of Fixed Order

Many of the commonly-used generating sets for the symmetric group Σn and the alternating group An either consist of a number of generators that grows as n grows or else contain an element whose order grows as n grows. For instance, Σn is generated by the set of adjacent transpositions, and it is also generated by a single transposition along with an n-cycle. Similarly, An is generated by all 3-...

Asymptotics of Generating the Symmetric and Alternating Groups

The probability that a random pair of elements from the alternating group An generates all of An is shown to have an asymptotic expansion of the form 1¡1/n¡ 1/n2¡4/n3¡23/n4¡171/n5¡ ... . This same asymptotic expansion is valid for the probability that a random pair of elements from the symmetric group Sn generates either An or Sn. Similar results hold for the case of r generators (r > 2). MSC 2...

Regular polytopes with symmetric and alternating groups

Generating infinite symmetric groups

Let S = Sym(Ω) be the group of all permutations of an infinite set Ω. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, respectively as a monoid, then there exists a positive integer n such that every element of S may be written as a group word, respectively a monoid word, of length ≤ n in the elements of U. Some related questions are n...

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.