Local compactness in free topological groups

We show that the subspace A(n)(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n is an element of omega if and only if A(2)(X) is locally compact if and only if F-2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace F-n(X) of the free topological group F(X) is locally compact for each n is an element of omega if and only if F-4(X) is locally compact if and only if F-n(X) has pointwise countable type for each n is an element of omega if and only if F-4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that A(n)(X) has pointwise countable type for each n is an element of omega if and only if A(2)(X) has pointwise countable type if and only if F-2(X) has pointwise countable type if and only if there exists a compact set C of countable chaxacter in X such that the complement X\C is discrete. Finally, we show that F-2(X) is locally compact if and only if F-3(X) is locally compact, and that F-2(X) has pointwise countable type if and only if F-3(X) has pointwise countable type.