More on convergent sequences in free topological groups

We show that, for every infinite cardinal τ, there exists a zero-dimensional pseudocompact space X   with |X|=τω|X|=τω such that all countable subsets of X are closed (hence all compact subsets of X   are finite), but both the free Abelian topological group A(X)A(X) and the free topological group F(X)F(X) on X contain a compact subspace of cardinality τ. This answers a recent question raised by A.V. Arhangelʼskii. We also show that there exists a countably compact space Y   without infinite compact subsets such that the groups F(Y)F(Y) and A(Y)A(Y) contain the one-point compactification of a discrete space of cardinality 2c2c (under additional set-theoretic assumptions, the groups F(Y)F(Y) and A(Y)A(Y) can contain arbitrarily big compact subsets).If, however, X   is homeomorphic to a topological group, then the existence of an infinite compact subset of F(X)F(X) (or A(X)A(X)) implies that X contains an infinite compact subset as well.