Intervals in subgroup lattices of infinite groups
An outstanding problem of the lattice theory asserts: is every finite lattice isomorphic to the congruence lattice of a finite algebra? \textit{P. P. Pálfy} and \textit{P. Pudlák} [Algebra Univers. 11, 22-27 (1980; Zbl 0386.06002)] proved that every finite lattice is isomorphic to the congruence lattice of a finite algebra if and only if every finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. The author deals with the infinite version of this theorem. The main result of the paper asserts: Every algebraic lattice is isomorphic to an interval in the subgroup lattice of an infinite group. The proof gives a new proof of the Grätzer-Schmidt theorem which characterizes the congruence lattices of algebras.
scientific article; zbMATH DE number 4112870