An important problem of the theory of Abelian groups is the study of the structure of their fully invariant subgroups and lattices formed by the subgroups. For some quite large classes of Abelian p-groups, a description of all fully invariant subgroups has been obtained in the works of Baer [6], Linton [41], Kaplansky [36], Benabdallah, Eisenstadt, Irvin, and Poluianov [9], and Pierce [51]. Little is known about fully invariant subgroups of Abelian torsion-free groups and mixed Abelian groups. The result of Gobel [27] on the structure of fully invariant subgroups of K-direct sums of infinite cyclic groups is known. In [42], Mader considers fully invariant and pure fully invariant subgroups of algebraically compact groups.
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