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Uncountable Homomorphic Images of Polish Groups are not ℵ 1 ‐Free Groups
Author(s) -
Khelif Anatole
Publication year - 2005
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/s0024609304003960
Subject(s) - uncountable set , mathematics , separable space , group (periodic table) , countable set , free group , homomorphic encryption , automorphism group , commutative property , image (mathematics) , combinatorics , automorphism , discrete mathematics , pure mathematics , artificial intelligence , computer science , mathematical analysis , encryption , operating system , chemistry , organic chemistry
Shelah has recently proved that an uncountable free group cannot be the automorphism group of a countable structure. In fact, he proved a more general result: an uncountable free group cannot be a Polish group. A natural question is: can an uncountable ℵ 1 ‐free group be a Polish group? A negative answer is given here; indeed, it is proved that an ℵ 1 ‐free group cannot be a homomorphic image of a Polish group. In fact, a stronger result is proved, involving a non‐commutative analogue of the notion of separable group. 2000 Mathematics Subject Classification 20E05.