Groups satisfying the weak chain conditions for normal subgroups (Q1093732)

The weak chain conditions of the title are defined as follows. A group G satisfies Max-\(\infty\) for normal subgroups, if and only if, in any strictly ascending chain \(G_ 1<G_ 2<..\). of normal subgroups of G, only finitely many indices \(| G_{i+1}:G_ i|\) are infinite. The condition Min-\(\infty\) for normal subgroups is defined similarly. These conditions in the context of locally nilpotent groups, have been studied by \textit{L. A. Kurdachenko} [Sib. Mat. Zh. 20, 1068-1076 (1979; Zbl 0425.20025)]. In the present paper, some of Kurdachenko's work is extended to certain locally soluble groups. A useful result is that Min- \(\infty\) and Max-\(\infty\) for normal subgroups are inherited by subgroups of finite index. The main result is Theorem B: Let G be a periodic locally soluble group with a uniform bound on the ranks of its chief factors. If G satisfies either Max-\(\infty\) or Min-\(\infty\) for normal subgroups, then G is Chernikov. Several corollaries are given.