Let X be a class of groups. A group G is called a minimal non-X-group if it is not an X-group but all of whose proper subgroups are X-groups. In [16], Xu proved that if G is a soluble minimal non-Baer-group, then G/G is a minimal nonnilpotent-group which possesses a maximal subgroup. In the present note, we prove that if G is a soluble minimal non-(finite-by-Baer)-group, then for all integer n ≥ 2, G/γ n (G ) is a minimal non-(finite-by-abelian)-group.

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.

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