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) F ( X n ) is topologically isomorphic to a subgroup of F ( X ) provided X is a compact space or, more generally, a kω 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 N C ω -spaces that includes, in particular, the k_ω k ω -spaces and the well-ordered spaces of ordinals [0, α ) [ 0 , α ) , for every ordinal α 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) F ( Z 2 ) . In addition, our space Z is countably compact, separable, and its square Z² Z 2 is not pseudocompact.