On central automorphisms and inner automorphisms of MD -groups

A group [Formula: see text] with finite rank is called an [Formula: see text]-group if every upper central factor of [Formula: see text] is a direct product of an abelian group with the maximal condition on subgroups and a divisible abelian group. Let [Formula: see text] and [Formula: see text] be the inner automorphism group and the group of all central automorphisms of [Formula: see text] fixing the center [Formula: see text] elementwise, respectively. We characterize all [Formula: see text]-groups [Formula: see text] for which [Formula: see text].