Generating infinite symmetric groups

Details Generating infinite symmetric groups Generating infinite symmetric groups

2004-01-22

arxiv.org/abs/math/0401304v2

Bull. London Math. Soc. 38 (2006) 429-440

Mathematics

Group Theory

9 pages. See also http://math.berkeley.edu/~gbergman/papers To appear, J.London Math. Soc.. Main results as in original versio

Scientific paper

10.1112/S0024609305018308

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, 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 \leq n in the elements of U. Several related questions are noted, and a brief proof is given of a result of Ore's on commutators that is used in the proof of the above result.

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