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.
2020 Mathematics Subject Classification:
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.