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Generating Infinite Symmetric Groups
Author(s) -
Bergman George M.
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305018308
Subject(s) - george (robot) , mathematics , combinatorics , element (criminal law) , integer (computer science) , group (periodic table) , algebra over a field , discrete mathematics , computer science , pure mathematics , artificial intelligence , physics , law , quantum mechanics , political science , programming language
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).