On groups with many normal subgroups (Q6617662)

Let \(G\) be a group and \(\mathfrak{n}(G)\) be the lattice of normal subgroups of \(G\). A non-empty set \(\Omega\) of subgroups of \(G\) has \(G\)-normal deviation \(0\) if either \(\Omega \subseteq \mathfrak{n}(G)\) or the set \(\Omega \setminus \mathfrak{n}(G)\) satisfies the minimal condition; if \(\delta > 0\) is any ordinal, has \(G\)-normal deviation \(\delta\) if for every descending chain \(H_{1} > H_{2} > \dots > H_{n} > \cdots\) of elements of the set \(\Omega \setminus \mathfrak{n}(G)\) there exists a positive integer \(t\) such that the interval \([H_{n}/H_{n+1}]\) has \(G\)-normal deviation strictly smaller than \(\delta\) for each \(n \geq t\), and \(\delta\) is the smallest ordinal with such a property. In particular, \(G\) has normal deviation if the lattice \(\mathfrak{L}(G)\) of all subgroups of \(G\) has \(G\)-normal deviation.\N\NThe main result in the paper under review is Theorem 1.1: Let \(G\) be a radical group with normal deviation. Then, either \(G\) is minimax or all its subgroups are normal. In particular, if \(G\) satisfies the minimal condition on non-normal subgroups, then either \(G\) is Černikov or all its subgroups are normal.