Strong Sperner property of the subgroup lattice of an Abelian \(p\)-group (Q1589772)

Let \(P\) be a graded poset. If for all \(m\), the cardinality of a subset of \(P\) (which does not contain a chain of length \(m+1\)) does not exceed the cardinality of the union of the \(m\) largest ranks of \(P\) then we say \(P\) is strongly Sperner. In this paper the authors use some technical lemmas to prove that for given \(n\) and \(k\) and a large enough prime, \(p\), the subgroup lattice \(L_{(k^1)}(p)\) of the Abelian \(p\)-group \((\mathbb{Z}/p^k\mathbb{Z})^n\) is strongly Sperner.