Groups with the weak minimal condition for non-subnormal subgroups
A descending chain \(H_1>H_2>\cdots\) of subgroups of a group \(G\) is called a descending \(\infty\)-chain if all the indices \(|H_n:H_{n+1}|\) are infinite. A group \(G\) is said to satisfying the weak minimal condition on non-subnormal subgroups if it has no infinite descending \(\infty\)-chains consisting of non-subnormal subgroups. Recall also that a group \(G\) is generalized radical if it has an ascending normal series whose factors are locally (nilpotent-by-finite). The authors prove that, if \(G\) is a generalized radical group satisfying the weak minimal condition on non-subnormal subgroups, then either \(G\) is a soluble-by-finite minimax group or all subgroups of \(G\) are subnormal.
weak minimal condition on non-subnormal subgroups
scientific article; zbMATH DE number 1383228
scientific article