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Introduction: In 2001, the outstanding mathematician Laszlo Fuchs et al. studied in Commun. Algebra the so-called  fully invariant extending property for abelian groups and modules, which concept arises quite naturally from the well-known classical definition of a fully invariant subgroup.

Definition 1. A subgroup H of an abelian group G is said to be fully invariant if f(H) < H for any endomorphism f : G \to G.

This can be curiously strengthened to the following notion.

Definition 2. A subgroup S of an abelian group G is said to be strongly invariant if \phi(S) < S for any homomorphism \phi : S\to G.

Clearly, strongly invariant subgroups are fully invariant as well as fully invariant direct summands are strongly invariant subgroups.

We thus come to

Main Definition. We shall say that a group G possesses the strongly invariant extending property (si-extending property for short) if every strongly invariant subgroup is contained as an essential subgroup in a direct summand of G.

Results: The following two statements are proved.

Theorem 1. A direct summand of a group having the si-extending property is also a group with the si-extending property.

Theorem 2. A group G has the si-extending property if, and only if, G = A\oplus B \oplus C, where A is a torsion group, B is a torsion-free group and C is a divisible group, each of which retains the si-extending property. 

The used technique for proofs exploits homological algebra and some variations of set theory.

This is a joint work with A.R. Chekhlov (Tomsk State Univ., Russia) and will be published in Quaestiones Mathematicae (2019).

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