Constructing Tychonoff G-spaces which are not G-Tychonoff

Jan de Vriesʹ compactification problem is whether every Tychonoff G-space can be equivariantly embedded in a compact G-space. In such a case, we say that G is a V-group. De Vries showed that every locally compact group G is a V-group. The first example of a non-V-group was constructed in 1988 by the first author. Until now, this was the only known counterexample. In this paper, we give a systematic method of constructing noncompactifiable G-spaces. We show that the class of non-V-groups is large and contains all second countable (even ℵ0-bounded) nonlocally precompact groups. This establishes the existence of monothetic (even cyclic) non-V-groups, answering a question of the first author. As a related result, we obtain a characterization of locally compact groups in terms of “G-normality”.