Hall chains in normal subgroups of finite p-groups

Some suffcient conditions for existence of k-admissible Hall chains (= \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{H}_k $$\end{document}-chains) in normal subgroups of finite p-groups are established (for irregular p-groups we consider only the case k = p). In Propositions 13–15 we study p-groups without \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{H}_p $$\end{document}-chains, and metacyclic 2-groups with the above property are classified. Abelian p-groups with exactly one \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{H}_k $$\end{document}-chain are characterized in Proposition 12.