The finite lattice representation problem and intervals in subgroup lattices of finite groups

A well-known result of universal algebra states: every algebraic lattice is isomorphic to the congruence lattice of an algebra. It is natural to ask whether every finite lattice occurs as the congruence lattice of a finite algebra. This fundamental question, asked over 45 years ago, is among the most elusive problems of universal algebra. Call a finite lattice L “representable” if it is the congruence lattice of a finite algebra. Pudlák and Tůma proved in 1980 that any finite lattice can be “concretely represented” as a sublattice of Eq(X), the equivalences of a finite set X . In the same year, Pálfy and Pudlák proved that the finite lattice representation problem is equivalent to a problem about lattices of subgroups of finite groups. In particular, they showed that the following statements are equivalent: (i) Any finite lattice is isomorphic to the congruence lattice of a finite algebra. (ii) Any finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. We discuss this remarkable result and the key ingredients of its proof.