Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- An army of cohomology against residual finiteness
- On some questions concerning subnormally monomial groups
- A conjecture concerning the evaluation of products of class-sums of the symmetric group
- Automorphisms of Burnside rings
- On finite generation of unit groups for group rings
- Counting finite index subgroups
- The quantum double of a finite group and its role in conformal field theory
- Closure properties of supersoluble Fitting classes
- Groups acting on locally finite graphs - a survey of the infinitely ended case
- An invitation to computational group theory
- On subgroups, transversals and commutators
- Intervals in subgroup lattices of finite groups
- Amalgams of minimal local subgroups and sporadic simple groups
- Vanishing orbit sums in group algebras of p-groups
- From stable equivalences to Rickard equivalences for blocks with cyclic defect
- Factorizations in which the factors have relatively prime orders
- Some problems and results in the theory of pro-p groups
- On equations in finite groups and invariants of subgroups
- Group presentations where the relators are proper powers
- A condensing theorem
- Lie methods in group theory
- Some new results on arithmetical problems in the theory of finite groups
- Groups that admit partial power automorphisms
- Problems
Intervals in subgroup lattices of finite groups
Published online by Cambridge University Press: 19 February 2010
- Frontmatter
- Contents
- Preface
- Introduction
- An army of cohomology against residual finiteness
- On some questions concerning subnormally monomial groups
- A conjecture concerning the evaluation of products of class-sums of the symmetric group
- Automorphisms of Burnside rings
- On finite generation of unit groups for group rings
- Counting finite index subgroups
- The quantum double of a finite group and its role in conformal field theory
- Closure properties of supersoluble Fitting classes
- Groups acting on locally finite graphs - a survey of the infinitely ended case
- An invitation to computational group theory
- On subgroups, transversals and commutators
- Intervals in subgroup lattices of finite groups
- Amalgams of minimal local subgroups and sporadic simple groups
- Vanishing orbit sums in group algebras of p-groups
- From stable equivalences to Rickard equivalences for blocks with cyclic defect
- Factorizations in which the factors have relatively prime orders
- Some problems and results in the theory of pro-p groups
- On equations in finite groups and invariants of subgroups
- Group presentations where the relators are proper powers
- A condensing theorem
- Lie methods in group theory
- Some new results on arithmetical problems in the theory of finite groups
- Groups that admit partial power automorphisms
- Problems
Summary
Introduction
In this survey paper we shall consider two closely related lattice representation problems, one from group theory, another from universal algebra. We shall put more emphasis on the group theoretical approach; an excellent survey with more universal algebraic flavour has been written by Thomas Ihringer [7].
Various problems concerning subgroup lattices have been studied already since the thirties, see the monograph of Michio Suzuki [25]. Our question deals with a relatively new aspect of this area; it concerns intervals in subgroup lattices. By an interval [H; G] in the subgroup lattice of a group G we mean the lattice of all subgroups of G containing the given subgroup H. The motivation for the following main problem will be given below.
Problem 1.1. Which finite lattices can be represented as intervals in subgroup lattices of finite groups?
Historically the problem originates from universal algebra, from the description of congruence lattices of algebraic structures. (For concepts from universal algebra see [12]; we shall also give a somewhat detailed discussion of the universal algebraic background in Section 7.) One of the first highly sophisticated construction techniques in this field was developed by G. Grätzer and E. T. Schmidt in their fundamental paper [4].
Theorem 1.2. (Grätzer and Schmidt [4]) Every algebraic lattice is isomorphic to the congruence lattice of some algebra.
Since every finite lattice is algebraic, every finite lattice can be represented as a congruence lattice of an algebra. However, the original construction in [4], as well as the more recent ones by Pudlák [18] and Tůma [27] all yield infinite algebras even if the lattice is finite.
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- Groups '93 Galway/St Andrews , pp. 482 - 494Publisher: Cambridge University PressPrint publication year: 1995
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