On the Socles of Fully Inert Subgroups of Abelian p-Groups

We define the so-called fully inert socle-regular and weakly fully inert socle-regular Abelian p-groups and study them with respect to certain of their numerous interesting properties. For instance, we prove that in the case of groups of length ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}, these two group classes coincide, but that in the case of groups of length ω+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega +1$$\end{document}, they differ. Some structural and characterization results are also obtained. The work generalizes concepts which have been of interest recently in the theory of entropy in algebra and builds on recent investigations by Danchev and Goldsmith (Arch Math (3) 92:191–199, 2009; J Algebra 323:3020–3028, 2010).