Completely inert subgroups of abelian groups

Peter Danchev; Andrey R. Chekhlov

doi:10.24193/subbmath.2026.1.01

Authors

Peter Danchev Institute of Mathematics and Informatics, Section of Algebra, Bulgarian Academy of Sciences, Sofia, Bulgaria https://orcid.org/0000-0002-2016-2336
Andrey R. Chekhlov Faculty of Mathematics and Mechanics, Section of Algebra, Tomsk State University, Russia https://orcid.org/0000-0002-9078-128X

DOI:

https://doi.org/10.24193/subbmath.2026.1.01

Keywords:

abelian groups, inert subgroups

Abstract

We define and study in-depth the so-called completely inert and uniformly completely inert subgroups of Abelian groups. We curiously show that a subgroup is completely inert exactly when it is characteristically inert. Moreover, we prove that a subgroup is uniformly completely inert precisely when it is uniformly characteristically inert. These two statements somewhat strengthen recent results due to Goldsmith-Salce established for totally inert subgroups in J. Commut. Algebra (2025). Some other closely relevant things are obtained as well.

References

[1] Breaz, S., Calugareanu, G., Strongly inert subgroups of abelian groups, Rend. Sem. Mat. Univ. Padova 138(2017), 101-114.

[2] Calugareanu, G., Strongly invariant subgroups, Glasgow Math. J. 57(2015), no. 2, 431443.

[3] Chekhlov, A. R., On fully inert subgroups of completely decomposable groups, Math. Notes 101(2017), no. 2, 365-373.

[4] Chekhlov, A. R., Fully inert subgroups of completely decomposable groups, having homogeneous components of the final rank, Russian Math. (Izv. VUZ) 66(12)(2022), no. 12, 82-90.

[5] Chekhlov, A. R., Danchev, P. V., Solution to the uniformly fully inert subgroups problem for Abelian groups, Rocky Mount. J. Math. 55(2025), no. 6, 1575-1578.

[6] Chekhlov, A. R., Danchev, P. V., and Goldsmith, B., On the socles of fully inert subgroups of Abelian p-groups, Mediterr. J. Math. 18(2021), no. 3.

[7] Chekhlov, A. R., Danchev, P. V., and Goldsmith, B., On the socles of characteristically inert subgroups of Abelian p-groups, Forum Math. 33(2021), no. 4, 889-898.

[8] Chekhlov, A. K., Danchev, P. V., and Keef, P. W., A note on uniformly totally strongly inert subgroups of Abelian groups, J.

[9] Danchev, P. V., and Keef, P. W., Abelian p-groups with minimal characteristic inertia, Commun. Algebra 51(2023), no. 6, 2308-2320.

[10] Dardano, U., Dikranjan, D., and Rinauro, S., Inertial properties in groups, Intern. J. Group Theory 7(2018), no. 3, 17-62.

[11] Fuchs, L., Infinite Abelian Groups, vols. I & II, Acad. Press, New York and London, 1970, 1973.

[12] Goldsmith, B., and Salce, L., Abelian p-groups with minimal full inertia, Period. Math. Hungar. 85(2022), no. 1, 1-13.

[13] Goldsmith, B., and Salce, L., Totally inert subgroups of Abelian groups, J. Commut. Algebra 17(2025), no. 3, 341-352.

[14] Kaplansky, I., Infinite Abelian Groups, University of Michigan Press, Ann Arbor (1954 & 1969).

[15] Keef, P. W., Countably totally projective Abelian p-groups have minimal full inertia, J. Commut. Algebra 14(2022), no. 3, 427-442.

[16] Krylov, P.A., Mikhalev, A. V., and Tuganbaev, A. A., Endomorphism Rings of Abelian Groups, Dordrecht, Boston and London, Kluwer Acad. Publ., 2003.

[17] Salce, L. An overview of some new classes of Abelian p-groups, Adv. Group Theory Appl. 19(2024), no. 2, 155-175.