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We investigate free groups over sequential spaces. In particular, we show that the free -group and the free sequential group over a sequential space with unique limits coincide and, barred the trivial case, their sequential order is .
We consider the following notion of largeness for subgroups of . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of can be extended to a large free subgroup of , and, under Martin’s Axiom, any free subgroup of of cardinality less than can also be extended to a large free subgroup of . Finally, if Gₙ are countable groups, then...
It is proven that an infinite-dimensional Banach space (considered as an Abelian topological group) is not topologically isomorphic to a subgroup of a product of -compact (or more generally, -bounded) topological groups. This answers a question of M. Tkachenko.
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