Reflection principle characterizing groups in which unconditionally closed sets are algebraic

Details Reflection principle characterizing groups in which unconditionally closed sets are algebraic Reflection principle characterizing groups in which unconditionally closed sets are algebraic

2007-03-11

arxiv.org/abs/math/0703304v1

Journal of Group Theory, 11 (2008), no. 3, 421-442

Mathematics

Group Theory

14 pages

Scientific paper

10.1515/JGT.2008.025

We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63 years old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G.

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