Necessary and sufficient conditions under which isomorphism of endomorphism groups of abelian p-groups A and B implies isomorphism of the groups A and B have been obtained. This yields a solution of L. Fuchs's problem 41 from his monograph "Abelian groups", Budapest, 1966. The question when the group Hom(A,C) is zero is completely solved for the case when at least one of the abelian groups A, C is a torsion group. The question when the homomorphism group Hom(A,C) is zero was studied in the case when C is a homogeneous separable group. New classes of fully transitive abelian groups for which fully invariant subgroups and their lattices are described have been discovered. This makes it possible to obtain results for known classes as corollaries. Fully invariant subgroups and their lattices are completely described for separable p-groups, separable torsion free groups, vector groups and mixed completely decomposable groups. Abelian groups from different classes in which the lattice of fully invariant subgroups is distributive or generally distributive are described. We studied f.i.-correct groups, i.e. groups for which an analog of the well-known set-theoretical Cantor-Schroder-Bernstein theorem is valid. (An abelian group A is said to be f.i.-correct if for any group B isomorphism of the group A and B follows from the fact that each of the group A, B is isomorphic to a fully invariant subgroup of the other group). f.i.-correct groups are completely described for different classes of torsion groups, torsion free groups amd mixed groups. Abelian p-groups with elements of infinite height have been studied. A wide class of such groups that yields a negative solution of I. Kaplansky's problem 25 from "Problems on Abelian groups", Proceedings of the Symposium of abelian groups, New Mexico, 1963, p. 365–368 is presented.
Main publications:
Grinshpon S. Ya. Primarnye abelevy gruppy s izomorfnymi gruppami endomorfizmov // Matem. zametki. 1973. T. 14, # 5. S. 733–741.
Grinshpon S. Ya. O ravenstve nulyu gruppy gomomorfizmov abelevykh grupp // Izv. vuzov. Matematika. 1998. # 9. S. 42–46.
Grinshpon S. Ya. Vpolne kharakteristicheskie podgruppy separabelnykh abelevykh grupp // Fundament. i prikl. matem. 1998. T. 4, vyp. 4. S. 1281–1307.
Grinshpon S. Ya. f.i.-korrektnye abelevy gruppy // Uspekhi matem. nauk. 1999. T. 54, # 6. S. 155–156.
Grinshpon S. Ya. Vpolne kharakteristicheskie podgruppy abelevykh grupp bez krucheniya i ikh reshetki // Fundament. i prikl. matem. 2000. T. 6, vyp. 3. S. 747–759.
