We investigate free groups over sequential spaces. In particular, we show that the free $k$-group and the free sequential group over a sequential space with unique limits coincide and, barred the trivial case, their sequential order is ${\omega }_1$.

We consider the following notion of largeness for subgroups of $S_{\infty }$. 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 $S_{\infty }$ can be extended to a large free subgroup of $S_{\infty }$, and, under Martin’s Axiom, any free subgroup of $S_{\infty }$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty }$. 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 $\sigma$-compact (or more generally, $o$-bounded) topological groups. This answers a question of M. Tkachenko.