Groups and lattices

Abstract

Abstract In this survey paper we discuss some topics from the theory of subgroup lattices. After giving a general overview, we investigate the local structure of subgroup lattices. A major open problem asks if every finite lattice occurs as an interval in the subgroup lattice of a finite group. Next we investigate laws that are valid in normal subgroup lattices. Then we sketch the proof that every finite distributive lattice is the normal subgroup lattice of a suitable finite solvable group. Finally, we discuss how far the subgroup lattice of a direct power of a finite group can determine the group. Introduction This survey paper is the written version of my four talks given at the Groups – St Andrews 2001 in Oxford conference. I selected some topics on subgroup lattices and normal subgroup lattices according to my personal taste and interest. These topics, of course, cannot cover all interesting and important parts of the theory. For a more complete overview the reader should consult the small book of Michio Suzuki [60] from 1956 and the more recent monograph by Roland Schmidt [54]. The latter one is a thick volume of 541 pages including 384 references. So it is clearly impossible to give a comprehensive survey here. My choice of topics was partly guided by the review of Schmidt's book by Ralph Freese [13].