Mathematics and Statistics Vol. 14(1), pp. 40 - 46
DOI: 10.13189/ms.2026.140103
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An Extended Construction of Hopfian Free-torsion Abelian Groups

Abderrahim Bouzendaga ^*, Seddik Abdelalim
Laboratory of Fundamental and Applied Mathematics (LMFA), Faculty of Sciences Ain Chock (FSAC), University Hassan II of Casablanca (UNIVH2C), Morocco

ABSTRACT

The study of Hopficity in Abelian groups has been largely motivated by the fundamental results of Baumslag, who proved that torsion groups are always Hopfian regardless of their cardinality, but left several questions open concerning torsion-free groups. Later, Corner addressed some of these questions by providing counterexamples showing that a direct sum of two Hopfian groups can be non-Hopfian, and that a group with an automorphism group of order two does not guarantee Hopficity. These results highlighted the need for new constructions to explore Hopficity in torsion-free Abelian groups. Our work introduces a new approach based on divisibility techniques, as it contributes to the understanding of free-torsion groups with respect to the Hopficity property, providing new insights into their structural properties and implications within group theory. Our analysis also demonstrates how divisibility properties, as well as the introduction of totally invariant subgroups and homomorphisms, can be used to establish Hopficity in specific Abelian groups, particularly those that are free-torsion. In order to reach all of this, we start by taking a group defined as an infinite direct sum of cyclic groups; then we construct a specific subgroup generated by two particular families of elements; and finally we show that this group is Hopfian through results from the theory of divisible subgroups.

KEYWORDS
Hopfian Groups, Free Torsion Groups, Abelian Groups, -divisible Groups, Maximal Divisible Groups, Fully Invariant Subgroups

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Abderrahim Bouzendaga , Seddik Abdelalim , "An Extended Construction of Hopfian Free-torsion Abelian Groups," Mathematics and Statistics, Vol. 14, No. 1, pp. 40 - 46, 2026. DOI: 10.13189/ms.2026.140103.

(b). APA Format:
Abderrahim Bouzendaga , Seddik Abdelalim (2026). An Extended Construction of Hopfian Free-torsion Abelian Groups. Mathematics and Statistics, 14(1), 40 - 46. DOI: 10.13189/ms.2026.140103.