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Generating infinite symmetric groups. (English) Zbl 1103.20003

H. D. Macpherson and P. M. Neumann [in J. Lond. Math. Soc., II. Ser. 42, No. 1, 64-84 (1990; Zbl 0668.20005)] showed that if \(\Omega\) is an infinite set, then the group \(S=\text{Sym}(\Omega)\) is not the union of a chain of \(|\Omega|\) or fewer proper subgroups. In this paper the author repeats the proof of that result, with modifications that allow us to obtain, along with it, the result stated in the form: 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 Ø. Ore’s on commutators [Proc. Am. Math. Soc. 2, 307-314 (1951; Zbl 0043.02402)], which is used in the proof of the above result.

MSC:

20B30 Symmetric groups
20B07 General theory for infinite permutation groups
20F05 Generators, relations, and presentations of groups
20M05 Free semigroups, generators and relations, word problems
20E15 Chains and lattices of subgroups, subnormal subgroups
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