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Bulletin of the London Mathematical Society 2006 38(3):429-440; doi:10.1112/S0024609305018308
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© London Mathematical Society

Generating Infinite Symmetric Groups

George M. Bergman

Department of Mathematics, University of California Berkeley, CA 94720-3840, USA; gbergman{at}math.berkeley.edu

Received 26 July 2004. Revision received 23 March 2005.

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. 2000 Mathematics Subject Classification 20B30 (primary), 20B07, 20E15 (secondary).


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