The author investigates groups with the minimal and maximal conditions on infinite subnormal subgroups (the min-\(\infty\)s and max-\(\infty\)s conditions, respectively). The groups with the minimal and the maximal conditions on subnormal subgroups (the min-s and max-s conditions, respectively) have been studied by D. J. S. Robinson [J. Algebra 10, 333-359 (1968; Zbl 0175.29802)] and L. A. Kurdachenko [Mat. Zametki 29, 19-30 (1981; Zbl 0471.20020)]. The author describes the soluble groups with the min-\(\infty\)s and max-\(\infty\)s conditions.
As an example we mention the following one of the main results. Theorem 3.1. A solvable group \(G\) satisfies max-\(\infty\)s if and only if either it is polycyclic or it is Prüfer-by-finite-by-finitely generated Abelian and the centralizer of the Prüfer subgroup of \(G\) is torsion.
The article contains many details about the structure of soluble groups with the min-\(\infty\)s and max-\(\infty\)s conditions.