Weak minimal condition for nonnilpotent subgroups (Q1061845)

The results in this paper are: (Theorem 1). Suppose G is a non-Chernikov locally finite group in which every infinite subgroup of infinite index is either nilpotent of class at most s or Chernikov. Then G is nilpotent of class at most s. (Corollary). Suppose G is a locally finite group which is not nilpotent of class at most s and in which every chain \(G_ 1>G_ 2>...>G_ n>..\). of subgroups is finite if (a) all the indices \([G_ n:G_{n+1}]\) are infinite and (b) the subgroups are not nilpotent of class at most s. Then G is Chernikov. (Theorem 2) Suppose G is a binary-finite (i.e., every 2-generator subgroup is finite) group in which every infinite subgroup of infinite index is nilpotent or Chernikov. Then G is locally finite. (Corollary) Suppose G is a binary-finite group in which each chain of non-nilpotent subgroups \(G_ 1>G_ 2>...G_ n>..\). is finite if all the indices \([G_ n:G_{n+1}]\) are infinite. Then G is locally finite. (Theorem 3) Suppose G is a non-Chernikov binary-finite group in which every infinite subgroup of infinite index is either nilpotent of class at most s or Chernikov. Then G is nilpotent of class at most s.