Subgroup Lattices of Finite Groups

By the fundamental theorem of Galois theory the lattice of subfields of a finite Galois extension is dually isomorphic to the subgroup lattice of the Galois group of the extension. So studying subgroup lattices is important not only for group theory, but also for its consequences in other fields. If the extension is not a Galois extension then the lattice of subfields is isomorphic to an interval in the subgroup lattice of the Galois group of the splitting field. Th e problem whether every finite lattice can be represented as an interval in the subgroup lattice of a finite group is still open, although it was formulated in 1980. Th is has relevance also for the problem on congruence lattices of finite algebras in universal algebra, as well as for the problem on intermediate subfactor lattices in functional analysis. Michael Aschbacher has recently published seven papers related to this problem. In his analysis of the question two types of groups play a prominent role: almost simple groups and certain twisted wreath products. In the talk the twisted wreath products - a construction introduced by B. H. Neumann in 1963 - will be explained in detail, yielding new examples of intervals in subgroup lattices.