Embedding the free topological group F(Xn) into F(X)

In 1976, Nickolas showed that for each natural n, the free topological group F(Xⁿ) is topologically isomorphic to a subgroup of F(X) provided X is a compact space or, more generally, a k_ω-space. We complement the Nickolas’ embedding theorem by showing that it remains true for every topological space X such that all finite powers of X are pseudocompact. For example, all pseudocompact k-spaces enjoy this property. Also, we extend the embedding theorem to the class of NC_ω-spaces that includes, in particular, the k_ω-spaces and the well-ordered spaces of ordinals [0,α), for every ordinal α. Our results are quite sharp because we present a first example of a Tychonoff space Z such that F(Z) does not contain an isomorphic copy of the group F(Z²). In addition, our space Z is countably compact, separable, and its square Z² is not pseudocompact.