Hall chains in normal subgroups of finite \(p\)-groups. (Q1001416)

Let \(G\) be a finite \(p\)-group, and \(H\) a normal subgroup of \(G\). A \(k\)-admissible chain in \(H\) is a chain \(\{1\}=L_0<L_1<\cdots<L_n=H\) of \(G\)-invariant subgroups such that each quotient group \(L_i/L_{i-1}\) has order at most \(p^k\) and exponent \(p\). Various questions about the existence of such chains are studied. In particular, Abelian \(p\)-groups with exactly one such chain are characterised, and metacyclic 2-groups without any such chain for \(k=2\) are classified.