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Generating infinite symmetric groups

George M. Bergman
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Abstract

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 \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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George M. Bergman. 2005-05-27. Generating infinite symmetric groups. https://doi.org/10.1112/s0024609305018308

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