\(M_ 7\) as an interval in a subgroup lattice
This note is a contribution to the problem, whether every finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. The problem originates from universal algebra [\textit{P. P. Pálfy} and \textit{P. Pudlák}, Algebra Univers. 11, 22-27 (1980; Zbl 0386.06002)]. Let \(M_ n\) be the lattice of length two with n atoms, G a finite group and H a subgroup of G. Suppose that the interval between H and G is the subgroup lattice of G is isomorphic to \(M_ n\) and H contains no nontrivial normal subgroup of G. The author proves that if n-1 is not a prime power then G is subdirectly irreducible with nonabelian minimal normal subgroup. Examples are known only for \(n=7\) and \(n=11\) [\textit{W. Feit}, ibid. 17, 220-221 (1983)].
interval in the subgroup lattice of a finite group
scientific article; zbMATH DE number 3904821