Book contents
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Groups and lattices
Published online by Cambridge University Press: 15 December 2009
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Summary
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].
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- Groups St Andrews 2001 in Oxford , pp. 428 - 454Publisher: Cambridge University PressPrint publication year: 2003
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