Proof of a conjecture on the Sperner property of the subgroup lattice of an Abelian \(p\)-group (Q1272363)

Let \(n,k\) be positive integers and \(p\) be a prime. Let \(L_{(k^n)}(p)\) be the subgroup lattice of the abelian \(p\)-group \((\mathbb{Z}/p^k\mathbb{Z})\times\dots\times(\mathbb{Z}/p^k\mathbb{Z})\) (\(n\) times). Confirming a conjecture of Stanley, the author proves that each middle level of \(L_{(k^n)}(p)\) is a maximal-sized antichain in this lattice, i.e., the lattice has the Sperner property. In the proof he uses a quotient theorem of Kleitman, Edelberg, and Lubell and a shifting argument in the poset \(L(n,k)\) of all tuples of integers \((\lambda_1,\dots, \lambda_n)\) with \(0\leq \lambda_1\leq\dots\leq \lambda_n\leq \lambda_k\), ordered componentwise.