It is known that the product of (every set of) pseudocompact topological groups is pseudocompact, so it becomes natural to inquire whether the product of countably compact groups is countably compact. The first substantive response to this question was given by E. K. van Douwen [Trans. Am. Math. Soc. 262, 417-427 (1980; Zbl 0453.54006)], who in the axiom system [ZFC+MA] produced countably compact groups \(G\) and \(H\) such that \(G\times H\) is not countably compact. Van Douwen (loc. cit.) asserted but did not prove that one may arrange in addition that \(G=H\).
The present authors, working in the system [ZFC+MA\(_{\text{countable}}]\), find a group \(H\) with the properties described in the title of this article. (MA\(_{\text{countable}}\) is a weak form of MA, equivalent to the condition that the real line cannot be covered by fewer than \({\mathfrak c}\)-many nowhere dense sets.) The authors’ group \(H\) satisfies \(H=G+D\subseteq\{0,1\}^{\mathfrak c}\) with \(D\) countable and \(G\) \(\omega\)-bounded, so \(H\) (unlike van Douwen’s groups) has many convergent sequences.
It remains an unsolved problem whether groups of this kind can be defined in ZFC.