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

Abstract 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 k 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_ N C ω -spaces that includes, in particular, the k_ k ω -spaces and the well-ordered spaces of ordinals [0, ) [ 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) F (Z 2). In addition, our space Z is countably compact, separable, and its square Z² Z 2 is not pseudocompact.