Groups with the real chain condition on non-pronormal subgroups (Q6617661)

A poset has deviation if and only if it contains no sub-poset order isomorphic to the poset \(\mathbb{Q}\) of rational numbers with their usual ordering. In [\textit{F. de Giovanni} et al., J. Algebra 613, 32--45 (2023; Zbl 1507.20015)], it has been shown that if \(G\) is a radical group and the set of non-pronormal subgroups has deviation, then either \(G\) is minimax or all subgroups are pronormal.\N\NA poset with the same order type as \(\mathbb{R}\) is a \(\mathbb{R}\)-chain and a poset has the real chain condition if it contains no \(\mathbb{R}\)-chain as subposets. Generalized radical groups with real chain condition on subgroups or with a subgroup theoretical property was considered by the authors [J. Algebra 642, 451--469 (2024; Zbl 1535.20169)]. In this paper, along the same lines, they obtain the following result: Let \(G\) be a generalized radical group. Then \(G\) satisfies the real chain condition on non-pronormal subgroups if and only if either \(G\) is a soluble-by-finite minimax group or all subgroups of \(G\) are pronormal.