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
Amalgams of minimal local subgroups and sporadic simple 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
Let H be a finite group and let p be a prime number dividing |H|. For S ∈ SylpH, put Loc(S) = {NH(R) | 1 ≠ R ≤ S}. A p-local subgroup of H is a subgroup of H which is in Loc(S) for some S ∈ SylpH.
The modern era of p-local analysis (meaning the study of various collections of p-local subgroups) may be considered to have its origin in Thompson's thesis [T1] (see also [T2] and [T3]). There he introduced what is now known as the Thompson order and subsequent refinements of this idea were important in the proof of the Odd Order Theorem [FT]. Questions concerning p-local subgroups frequently play a role in important problems about finite groups. This has been particularly true of the work which resulted in the classification of the finite simple groups.
In 1980 Goldschmidt inaugurated the amalgam method when in [Gol] he used amalgams to study a configuration arising in the N-group paper. More recently the amalgam method has been deployed in the revision of the simple group classification (see, for example, [S1]).
The basic situation in which the amalgam method applies is given in
Hypothesis 1.1.G is a group containing distinct finite subgroups P1 and P2 such that
(i) G = 〈P1, P2〉; and
(ii) P1 ∪ P2 contains no non-trivial normal subgroups of G.
- Type
- Chapter
- Information
- Groups '93 Galway/St Andrews , pp. 495 - 506Publisher: Cambridge University PressPrint publication year: 1995