Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank

A group G has finite rank r if every finitely generated subgroup of G is at most r-generator. If C is a class of groups then we let C* denote the class of groups G in which every proper subgroup of G is either of finite rank or in C. We let denote the class of soluble groups and the class of soluble groups of derived length at most d, where d is a positive integer. We let λ denote the set of closure operations and let denote the λ-closure of the class of periodic locally graded groups. Amongst other results we prove that a soluble -group is either of finite rank or of derived length at most d and also that a group in the class is either locally soluble, or has finite rank, or is isomorphic to one of or for suitable locally finite fields . The second author would like to thank the Department of Mathematics at Bucknell University for its hospitality while part of this work was being done.     

Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)

Our main result is that a locally graded group whose proper subgroups are Baer-by-Chernikov is itself Baer-by-Chernikov. We prove also that a locally (soluble-by-finite) group whose proper subgroups are Baer-by-(finite rank) is itself Baer-by-(finite rank) if either it is locally of finite rank but not locally finite or it has no infinite simple images.     

Groups with all subgroups permutable or of finite rank

In this paper we investigate the structure of X-groups in which every subgroup is permutable or of finite rank. We show that every subgroup of such a group is permutable.     

On groups with all proper subgroups finite-by-abelian-by-finite

Archiv der Mathematik - We show that if all proper subgroups of a locally graded group G are finite-by-abelian-by-finite, then G contains a finite normal subgroup N such that all proper subgroups...     

Groups with finite rank elements

With the aid of the notion of the rank of an element in an arbitrary group, we prove a criterion for an infinite group to be nonsimple and find conditions under which a q-biprimitively finite group G with Chernikov Sylow q-subgroups has a Chernikov quotient group G/O_p(G).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 836–839, June, 1992.     

On groups whose all proper subgroups have Chernikov derived subgroups

We study the groups for which the derived subgroups of all proper subgroups are Chernikov subgroups under condition that they possess a normal system whose factors are locally graduated.     

Groups with given properties of finite subgroups

Suppose that in every finite even order subgroup F of a periodic group G, the equality [u, x]² = 1 holds for any involution u of F and for an arbitrary element x of F. Then the subgroup I generated by all involutions in G is locally finite and is a 2-group. In addition, the normal closure of every subgroup of order 2 in G is commutative.     

Groups whose proper subgroups are Baer groups

In this paper we classify infinite soluble minimal non-nilpotent-groups, detemine the basic structure of infinite soluble minimal non-Baer-groups, and using famed Heineken-Mohamed-groups we construct an example of minimal non-Baer-group which is not minimal non-nilpotent-group.The author would like to thank Chen Zhangmu and Shi Wuje for narm help and useful advice.     

On 2-groups with finite subgroups of rank 2

A classification is achieved of the 2-groups all of whose finite subgroups are generated by two elements.