Pseudocompact and countably compact abelian groups: Cartesian products and minimality

Denote by $\mathcal{G}$ the class of all Abelian Hausdorff topological groups. A group $G \in \mathcal{G}$ is minimal (totally minimal) if every continuous group isomorphism (homomorphism) $i:G \to H$ of $G$ onto $H \in \mathcal{G}$ is open. For $G \in \mathcal{G}$ let $\kappa (G)$ be the smallest cardinal $\tau \geq 1$ such that the minimality of ${G^\tau }$ implies the minimality of all powers of $G$. For $\mathcal{Q} \subset \mathcal{G}$, $\mathcal{Q} \ne \emptyset$, we set $\kappa (\mathcal{Q}) = \sup \{ \kappa (G):G \in \mathcal{G}\}$ and denote by $\alpha (\mathcal{Q})$ the smallest cardinal $\tau \geq 1$ having the following property: If $\{ {G_i}:i \in I\} \subset \mathcal{Q}$, $I \ne \emptyset$, and each subproduct $\prod {\{ {G_i}:i \in J\} }$, with $J \subset I$, $J \ne \emptyset$, and $\vert J\vert \leq \tau$, is minimal, then the whole product $\prod {\{ {G_i}:i \in I\} }$ is minimal. These definitions are correct, and $\kappa (G) \leq {2^\omega }$ and $\kappa (\mathcal{Q}) \leq \alpha (\mathcal{Q}) \leq {2^\omega }$ for all $G \in \mathcal{G}$ and any $\mathcal{Q} \subset \mathcal{G}$, $\mathcal{Q} \ne \emptyset$, while it can happen that $\kappa (\mathcal{Q}) < \alpha (\mathcal{Q})$ for some $\mathcal{Q} \subset \mathcal{G}$. Let $\mathcal{C} = \{ G \in \mathcal{G}:G\;{\text{is}}\;{\text{countably}}\;{\text{compact}}\}$ and $\mathcal{P} = \{ G \in \mathcal{G}:G\;{\text{is}}\;{\text{pseudocompact}}\}$. If $G \in \mathcal{C}$ is minimal, then $G \times H$ is minimal for each minimal (not necessarily Abelian) group $H$; in particular, ${G^n}$ is minimal for every natural number $n$. We show that $\alpha (\mathcal{C}) = \omega$, and so either $\kappa (\mathcal{C}) = 1$ or $\kappa (\mathcal{C}) = \omega$. Under Lusin's Hypothesis ${2^{{\omega _1}}} = {2^\omega }$ we construct $\{ {G_n}:n \in \mathbb{N}\} \subset \mathcal{P}$ and $H \in \mathcal{P}$ such that: (i) whenever $n \in \mathbb{N}$, $G_n^n$ is totally minimal, but $G_n^{n + 1}$ is not even minimal, so $\kappa ({G_n}) = n + 1$; and (ii) ${H^n}$ is totally minimal for each natural number $n$, but ${H^\omega}$ is not even minimal, so $\kappa (H) = \omega$. Under ${\text{MA}} + \neg {\text{CH}}$, conjunction of Martin's Axiom with the negation of the Continuum Hypothesis, we construct $G \in \mathcal{P}$ such that ${G^\tau }$ is totally minimal for each $\tau < {2^\omega }$, while ${G^{{2^\omega }}}$ is not minimal, so $\kappa (G) = {2^\omega }$. This yields $\alpha (\mathcal{P}) = \kappa (\mathcal{P}) = {2^\omega }$ under ${\text{MA}} + \neg {\text{CH}}$. We also present an example of a noncompact minimal group $G \in \mathcal{C}$, which should be compared with the following result obtained by the authors quite recently: Totally minimal groups $G \in \mathcal{C}$ are compact.