Locally soluble-by-finite groups with the weak minimal condition on non-nilpotent subgroups (Q1614636)

Let \(G\) be a group and \(M\) a family of subgroups of \(G\). We say that \(G\) satisfies the condition Min-\(\infty\)-\(M\), the weak minimal condition for \(M\)-subgroups, if for every descending chain of \(M\)-subgroups \(H_1>H_2>\cdots>H_n>\cdots\) there is a number \(k\) such that \(|H_j:H_{j+1}|\) is finite for \(j\geq k\). Dually is defined the weak maximal condition for \(M\)-subgroups. The weak minimal and weak maximal conditions were introduced by R. Baer and D. I. Zaitsev in 1968. Groups with weak minimal or maximal conditions for distinct natural families of subgroups \(M\) (all subgroups, Abelian subgroups, non-Abelian subgroups, normal subgroups, non-normal subgroups, subnormal subgroups, non-subnormal subgroups) have been considered by several authors. If \(M\) is the family of all non-nilpotent subgroups, then Min-\(\infty\)-\(M\) is the weak minimal condition for non-nilpotent subgroups or, shortly, Min-\(\infty\)-(non-nil). The study of groups with Min-\(\infty\)-(non-nil) was started earlier by A. N. Ostylovskij, H. Smith, M. Dixon and M. Evans. In the present paper the authors continue these researches. The main results are the following. Theorem A. Let \(G\) be a locally (soluble-by-finite) group satisfying Min-\(\infty\)-(non-nil). Then (1) \(G\) contains a normal nilpotent subgroup \(H\) such that \(G/H\) is soluble-by-finite and minimax. (2) \(G\) is either minimax or locally nilpotent. Theorem B. Let \(G\) be a locally nilpotent group satisfying Min-\(\infty\)-(non-nil) and suppose that \(G\) is neither nilpotent nor minimax. Let \(K\) be the finite residual of \(G\). Then (1) \(G/K\) is minimax. (2) \(K\) is a periodic nilpotent subgroup with finite set \(\pi(K)\). (3) If \(L\) is a non-nilpotent and non-minimax subgroup of \(G\), then \(L\) contains \(K\).