Quasiprimitive groups and congruence lattices of finite algebras
by
Ferdinand Boerner
University of Potsdam, Germany

A transitive permutaion group is called quasiprimitive if all of its nontrivial normal subgroups are transitive. Similar as in the O´Nan-Scott Theorem for primitive groups, quasiprimitive groups can be classified according to the properties of their minimal normal subgroups (C. Praeger, 1993).

The finite lattice representation problem is the (open) question, whether every finite lattice is isomorphic to the congruence lattice of a finite algebra. This problem is equivalent to the question, whether every finite lattice is isomorphic to the subgroup lattice of a finite group (P.P. Palfy and P. Pudlak, 1980).

Let G be a transitive permutation group on a finite set A and let e in A. If now the interval [G_(e);G] in the subgroup lattice of G has a special property, called the LP-property, then G is quasiprimitive. On the other hand, the LP-property is relatively weak. Therefore many lattices have this property.

Using some ideas of R. Baddeley (1998), we investigate the structure of groups with special intervals in their subgroup lattice. Finally, we obtain an equivalent formulation of the finite lattice representation problem.

Date received: December 30, 2001