On groups that are nearly metahamiltonian

We consider groups whose subgroups are either normal or have Chernikov commutant. It was proved that if such group G has a subgroup H of finite index and its commutant is a Chernikov group, then commutant of group G is a Chernikov group.

Groups with finitely many non-isomorphic factor-groups

We prove that if [Formula: see text] is either a hypercentral-by-finite group or a soluble Baer group and if [Formula: see text] has finitely many non-isomorphic factor-groups, then [Formula: see text] is a Chernikov group. The converse is also true. Furthermore, we give some information on the structure of a metabelian group with finitely many non-isomorphic factor-groups.

Rank-Engel conditions on linear groups

We study linear groups G for which for every g in G there exists a subgroup E of G satisfying some sort of rank condition and such that for every x in G there is a positive integer m such that for all n ≥ m the repeated commutator [x, ng] lies in E. If E can always be chosen to be a Chernikov group (resp. a polycyclic-by-finite group) such G can be completely described (Wehrfritz in Adv Group Theory Appl 7:143–157, 2019; Wehrfritz in Boll Unione Mat Ital, to appear). For more general rank conditions our analyses below are complete for positive characteristics but are less so for characteristic zero.

On certain groups with Chernikov group of automorphisms

We obtained automorphic analogue of Schur’s theorem for the case when an arbitrary subgroup A of automorphism group Aut(G) of a group G and the factor-group of a group G modulo A-center are Chernikov groups.