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. Černikov 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 Černikov, satisfies the minimal condition on non-Baer subgroups, then it is a Baer-by-Černikov group which is a direct product of a p-subgroup containing a minimal non-Baer subgroup of infinite rank, by a Černikov 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.