On groups with the minimal condition for non-invariant decomposable abelian subgroups

The infinite groups, in which there is no any infinite descending chain of non-invariant decomposable abelian subgroups ($md$-groups) are studied. Infinite groups with the minimal condition for non-invariant abelian subgroups, infinite groups with the condition of normality for all decomposable abelian subgroups and others belong to the class of $md$-groups. The complete description of infinite locally finite and locally soluble non-periodic $md$-groups is given, the connection of the class of $md$-groups with other classes of groups are investigated.