Hall chains in normal subgroups of finite p-groups

Abstract

Some suffcient conditions for existence of k-admissible Hall chains (= \( \mathcal{H}_k \)-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 \( \mathcal{H}_p \)-chains, and metacyclic 2-groups with the above property are classified. Abelian p-groups with exactly one \( \mathcal{H}_k \)-chain are characterized in Proposition 12.