The amazing infinite symmetric group
by
George Bergman
Department of Mathematics, University of California at Berkeley

Suppose G is a group and X a subset of G. For i = 1, 2, ..., let X⁽ⁱ⁾ be the set of elements of G that can be written as words of length £ i in the elements of X. Thus, X generates G if and only if the union of the X⁽ⁱ⁾ is G; but, of course, in this situation no single X⁽ⁱ⁾ need equal G.

However, the full permutation group G on an infinite set W is an impatient animal; it will only allow itself to be generated by a set X if X⁽ⁱ⁾ = G for some i. I will sketch the proof of this and some related results, due (in various combinations) to D. Macpherson, P. Neumann, myself and S. Shelah. Details can be found in the two preprints on infinite symmetric groups at http://math.berkeley.edu/~gbergman/papers/ , and papers cited in these.

Date received: July 9, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoc-21.