Groups with some restrictions on non-Baer subgroups

Abstract

It is proved that if G is an $$\mathfrak {X}$$ -group of infinite rank whose proper subgroups of infinite rank are Baer groups, then so are all proper subgroups of G, where $$\mathfrak {X}$$ is the class defined by N.S. Cernikov as the closure of the class of periodic locally graded groups by the closure operations $$\varvec{\acute{P}}$$ , $$\varvec{\grave{P}}$$ and $$ \varvec{L}$$ . We prove also that if a locally graded group, which is neither Baer nor Cernikov, satisfies the minimal condition on non-Baer subgroups, then it is a Baer-by-Cernikov group which is a direct product of a p-subgroup containing a minimal non-Baer subgroup of infinite rank, by a Cernikov nilpotent $$p^{\prime }$$ -subgroup, for some prime p. Our last result states that a group is locally graded and has only finitely many conjugacy classes of non-Baer subgroups if, and only if, it is Baer-by-finite and has only finitely many non-Baer subgroups.