Generating Infinite Symmetric Groups

By George M. Bergman. 2004

108 citations recorded by the source corpus.

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, 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).

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/a65e52702a03d409084c39e943897533df098777

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Citations in the corpus

Distortion in transformation groups (2005, background): “Subsequently, intensive research on the subject has been carried on (see, among others [3, 4, 8, 9], and the references in [ 1 ]).”
Automatic continuity of homomorphisms and fixed points on metric compacta (2006, methodology): “✷ The result above implies that groups with Bergman’s property and with a comeager conjugacy class cannot have non-trivial representations by automorphisms of locally finite graphs.”
Generating automorphism groups of chains (2005, background): “The authors thank George Bergman and an anonymous referee for helpful comments made during the final preparation of this paper.”
Strongly Bounded Groups and Infinite Powers of Finite Groups (2004, background): “We observe that the class of discrete, strongly bounded groups coincides with a class of groups that has recently emerged since a preprint of Bergman (2005), sometimes referred as groups with “uncountable strong cofinality”, or “groups with uncountable cofinality and Bergman’s Property.”
A topological version of the Bergman property (2005, background): “In his paper [ 6 ], Bergman originally asked whether strong uncountable cofinality was preserved between a group and its subgroups of finite index, and A. Kh´ elif, in an announcement [22], stated that this is indeed the case.”
Property (T) and rigidity for actions on Banach spaces (2005, background): “As pointed out by Y. de Cornulier [Cr], uncountable discrete groups that have G. Bergman’s cofinality property [ Bn ] have (FH) but fail (T).) Some of the above applications use this latter characterization.”