Review of: Subgroup lattices of groups, by Roland Schmidt, Expositions in Math., vol. 14, de Gruyter, 1994, xv+572 pp.

Lattice theory originated late in the last century along two strands. Many of the structures used in logic have lattices, particularly distributive lattices, associated with them. The work of Boole [4] and Schröder [27] are seminal in this regard. The other strand, which concentrated more on the connections with algebra, was initiated with Richard Dedekind [6] in his studies of ideals in algebraic number fields. The lattices he studied usually satisfied his modular law which is sometimes referred to as Dedekind’s law. Lattice theory was used as a tool for deriving some of the basic structure theorems for group theory and for algebraic systems in general. In [19] and [20] O. Ore gave a purely lattice theoretic proof of the Krull-Schmidt theorem on the uniqueness of direct decompositions, considerably broadening the scope of this result. (See [18] for a clear lattice theoretic proof.) In [21] and [22] he applied lattices, which he called structures, to group theory. R. Baer, J. von Neumann and others employed techniques from lattice theory and projective geometry in proving theorems about groups and other algebraic structures, and Marshall Hall’s book on group theory contained chapters on both lattice theory and projective geometry. These results led to the hope that lattice theory might prove to be a powerful tool in group theory. In the introduction to his book [28] Suzuki concluded from his theorem that if G is a simple group, then G is determined by the lattice of subgroups ofG×G, that “we might have a possibility to apply lattice theoretical considerations to solve the classification problem of finite simple groups.” However this hope was not realized; much more powerful techniques, primarily character theory and “local analysis”, were used. Similarly in Abelian group theory Baer’s lattice theory techniques are no longer used. (See page 86 of Kaplansky’s monograph [13].) So group theory and lattice theory went their separate ways. (For that matter, group theory nowadays has little in common with Abelian group theory.) Group theory had other techniques and lattice theory had its own deep problems to work on, and most of the applications of lattice theory to algebra were in the field of universal algebra. In the last several years some connections between lattice theory and group theory have resurfaced. One problem of interest in general algebra: is every finite lattice isomorphic to the congruence lattice of some algebraic system? P. Pálfy and