0-1 embeddings of \(M_\ell\) in Abelian subgroup lattices. (Q2920249)

Denote by \(M_l\) the lattice with top, bottom and \(l\) incomparable atoms. A small step is made towards answering the following question: for which positive integers \(l\), does there exist a finite Abelian group \(G\) such that \(M_l\) embeds as a \(0\)-\(1\) sublattice in \(\mathbf{Sub}(G)\) (the subgroup lattice of \(G\)) but \(M_{l+1}\) does not?NEWLINENEWLINE More precisely, the following result is proved. Theorem. Let \(n\geq 2\) be a positive integer and let \(l\) be the largest integer such that \(M_l\) embeds as a \(0\)-\(1\) sublattice of \(\mathbf{Sub}(\mathbf Z_n\times\mathbf Z_n)\). Then \(l=p+1\), where \(p\) is the smallest prime number dividing \(n\). Furthermore, if \(M_l\) embeds but \(M_{l+1}\) does not, then the images of the atoms of \(M_l\) are all cyclic subgroups of order \(n\).