Embedding the free topological group \(F(X^n)\) into \(F(X)\) (Q6542386)

For a Tychonoff space \(X\), let \(F(X)\) denote the free topological group over \(X\). If \(X\) is a \(k_\omega\)-space, \textit{P. Nickolas} [J. Lond. Math. Soc., II. Ser. 12, 199--205 (1976; Zbl 0318.22002)] showed that the group \(F(X^n)\) is topologically isomorphic to a subgroup of \(F(X)\). \N\NIn this article, the authors extend Nickolas' embedding theorem proving that the same conclusion holds also for the following classes of Tychonoff spaces: \N\begin{enumerate}\N\item \(X\) is such that all finite powers of \(X\) are pseudocompact,\N\item \(X\) is an \(NC_\omega\)-space.\end{enumerate} \N\NOn the other hand, they construct a countably compact, separable space \(Z\) whose square \(Z^2\) is not pseudo-compact such that \(F(Z)\) does not contain an isomorphic copy of \(F(Z^2)\).