UC Berkeley

Let S = Sym(Omega) be the group of all permutations of an infinite set Omega. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, then there exists a positive integer n such that every element of S may be written as a group word of length at most n in the elements of U. Likewise, if U is a generating set for S as a monoid, then there exists a positive integer n such that every element of S may be written as a monoid word of length at most n in the elements of U. Some related questions and recent results are noted, and a brief proof is given of a result of Ore's on commutators, which is used in the proof of the above result.