Pseudocompactness and uniform continuity in topological groups

Summary:Let $\tau$ be an uncountable regular cardinal and $G$ a $T_1$ topological group. We prove the following statements: (1) If $\tau$ is homeomorphic to a closed subspace of $G$, $G$ is Abelian, and the order of every non-neutral element of $G$ is greater than $5$ then $\tau\times\tau$ embeds in $G$ as a closed subspace. (2) If $G$ is Abelian, algebraically generated by $\tau\subset G$, and the order of every element does not exceed $3$ then $\tau\times \tau$ is not embeddable in $G$. (3) There exists an Abelian topological group $H$ such that $\omega_1$ is homeomorphic to a closed subspace of $H$ and $\{t^2:t\in T\}$ is not closed in $H$ whenever $T\subset H$ is homeomorphic to $\omega_1$. Some other results are obtained