Introduction

In this survey paper we shall consider two closely related lattice representation problems, one from group theory, another from universal algebra. We shall put more emphasis on the group theoretical approach; an excellent survey with more universal algebraic flavour has been written by Thomas Ihringer [7].

Various problems concerning subgroup lattices have been studied already since the thirties, see the monograph of Michio Suzuki [25]. Our question deals with a relatively new aspect of this area; it concerns intervals in subgroup lattices. By an interval [H; G] in the subgroup lattice of a group G we mean the lattice of all subgroups of G containing the given subgroup H. The motivation for the following main problem will be given below.

Problem 1.1. Which finite lattices can be represented as intervals in subgroup lattices of finite groups?

Historically the problem originates from universal algebra, from the description of congruence lattices of algebraic structures. (For concepts from universal algebra see [12]; we shall also give a somewhat detailed discussion of the universal algebraic background in Section 7.) One of the first highly sophisticated construction techniques in this field was developed by G. Grätzer and E. T. Schmidt in their fundamental paper [4].

Theorem 1.2. (Grätzer and Schmidt [4]) Every algebraic lattice is isomorphic to the congruence lattice of some algebra.

Since every finite lattice is algebraic, every finite lattice can be represented as a congruence lattice of an algebra. However, the original construction in [4], as well as the more recent ones by Pudlák [18] and Tůma [27] all yield infinite algebras even if the lattice is finite.