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Reflection principle characterizing groups in which unconditionally closed sets are algebraic
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Dikran Dikranjan
Published/Copyright:
May 21, 2008
Abstract
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-year-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. Essential connections with non-topologizable groups are highlighted.
Received: 2007-03-11
Published Online: 2008-05-21
Published in Print: 2008-May
© de Gruyter 2008
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- Involution models of finite Coxeter groups
- Prime divisors of character degrees
- A characterization of HN
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- Conjugacy classes of non-normal subgroups in finite nilpotent groups
- Normalizers of subgroups of division rings
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Articles in the same Issue
- Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents
- Involution models of finite Coxeter groups
- Prime divisors of character degrees
- A characterization of HN
- Symmetric groups and conjugacy classes
- Conjugacy classes of non-normal subgroups in finite nilpotent groups
- Normalizers of subgroups of division rings
- On the Frattini and upper near Frattini subgroups of a generalized free product
- Reflection principle characterizing groups in which unconditionally closed sets are algebraic
