Anti Boolean Conditions for sets of subgroups of a group, preprint version, january 2025

We show that, in the universe of generalised soluble groups and for the most common generalisations \chi of normality, the condition that in the set of non-\chi subgroups of a group there are no copies of the boolean algebra of subsets of a countable set guarantiees that the structure of the group is under control. This incorporates many previuos results from the wider literature on the topic. Moreover, as an application, we point out how some specific antichains control the cardinality of the whole set of subgroups.