Theorem on groups of finite special rank

Locally (Soluble-by-Finite) Groups With Various Restrictions on Subgroups of Infinite Rank

“…(c) A locally finite group of finite rank is almost locally soluble, that is, it has a normal locally soluble subgroup of finite index [14]. Indeed, the same holds for locally (soluble-by-finite) groups, by [3] (see also [4]).…”

“…Clearly there is an integer r such that every finitely generated subgroup of G is r-generated, that is, G has rank at most r. By [3] G is almost locally soluble and hence almost soluble, since there is a finite upper bound for the derived length of soluble subgroups. Since our hypothesis is inherited by homomorphic images we may assume by induction that G is abelian-by-finite.…”

Locally Soluble-by-Finite Groups with the Weak Minimal Condition on Non-Nilpotent Subgroups

“…In [1],Černikov introduced an extensive class of groups and proved that -groups of finite rank are almost locally soluble. Since all locally soluble-by-finite groups belong to ,Černikov's result and Lemma 7 together show that G is almost locally soluble and of finite rank.…”

Groups with normality conditions for subgroups of infinite rank

“…Since the hypotheses are inherited by subgroups, we may choose a finitely generated counterexample G. It follows from a result of N. S.Černikov [2] that G has infinite rank. Let L be a chain of normal subgroups of G such that for every L ∈ L the factor group G/L is not soluble-by-finite, and put K =…”