Weakly fully and characteristically inert socle-regular Abelian p-groups

Abstract

In regard to two recent publications in the Mediterranean J. Math. (2021) and Forum Math. (2021) related to fully and characteristically inert socle-regularity, respectively, we define and study the so-called weakly characteristically inert socle-regular groups. In that aspect, as a culmination of the investigations of this sort, some more global results are obtained and, moreover, some new concrete results concerning the weakly fully inert socle-regular groups, defined as in the firstly mentioned above paper, are also established. In particular, we prove that all torsion-complete groups are characteristically inert socle-regular, which encompasses an achievement from the secondly mentioned paper and completely settles the problem posed there about this class of groups.

Keywords:

• Fully (characteristically) inert subgroups
• fully (characteristically) inert socle-regular groups
• socle-regular groups
• weakly fully (weakly characteristically) inert socle-regular groups

2020 Mathematics Subject Classification:

• 20K10

Acknowledgements

The authors are very grateful to their colleague and co-author Brendan Goldsmith from Dublin Technological Institute for his careful reading of the text and the constructive comments and suggestions on both presentation and proofs made. They are also very appreciated to the anonymous expert referee for his/her quite well-systematic, helpful and insightful report given.

Additional information

Funding

The work of the first-named author, A.R. Chekhlov, was supported by the Ministry of Science and Higher Education of Russia (agreement No. 075-02-2022-884). The work of the second-named author, P.V. Danchev, was partially supported by the Bulgarian National Science Fund under Grant KP-06 No. 32/1 of December 07, 2019.