Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups

I t is well-known that every algebraic lattice is isomorphic to the congruence lattice of an algebra. In this pape r we are interested in the prob lem of characterizing the finite lattices, which are isomorphic to the congruence lattices of finite algebras. We are not able to settle the prob lem if every finite lattice is isomorphic to the congruence lattice of a finite algebra (cf. [1], Problem 13). Our main result shows that this problem is related to the prob lem to characterize intervals in subgroup lattices of finite groups. Namely, 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.