Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Geometry, Steinberg representations and complexity
- The structure of metabelian finite groups
- Table algebras of extended Gagola-type and applications to finite group theory
- On the saturation of formations of finite groups
- Locally constructed formations of finite groups
- Reflections on virtually one-relator groups
- Rickard equivalences and block theory
- Computing the conjugacy classes of elements of a finite group
- Quotient categories of modules over group algebras
- Weak chain conditions for non-almost normal subgroups
- Computation of the character table of affine groups using Fischer matrices
- The lattice of compact representations of an infinite group
- Automorphisms of nilpotent and related groups
- Generation of orthogonal groups over finite fields
- The structure of certain Coxeter groups
- n-free groups and questions about universally free groups
- Classification of all generating pairs of two generator Fuchsian groups
- Parametric words and models of the elementary theory of non-abelian free groups
- The groups G(n, l) as fundamental groups of Seifert fibered homology spheres
- Lifting automorphisms: a survey
- (MI)-groups acting uniserially on a normal subgroup
- Revisiting a theorem of Higman
- Cohomological finiteness conditions
Rickard equivalences and block theory
Published online by Cambridge University Press: 02 March 2010
- Frontmatter
- Contents
- Preface
- Introduction
- Geometry, Steinberg representations and complexity
- The structure of metabelian finite groups
- Table algebras of extended Gagola-type and applications to finite group theory
- On the saturation of formations of finite groups
- Locally constructed formations of finite groups
- Reflections on virtually one-relator groups
- Rickard equivalences and block theory
- Computing the conjugacy classes of elements of a finite group
- Quotient categories of modules over group algebras
- Weak chain conditions for non-almost normal subgroups
- Computation of the character table of affine groups using Fischer matrices
- The lattice of compact representations of an infinite group
- Automorphisms of nilpotent and related groups
- Generation of orthogonal groups over finite fields
- The structure of certain Coxeter groups
- n-free groups and questions about universally free groups
- Classification of all generating pairs of two generator Fuchsian groups
- Parametric words and models of the elementary theory of non-abelian free groups
- The groups G(n, l) as fundamental groups of Seifert fibered homology spheres
- Lifting automorphisms: a survey
- (MI)-groups acting uniserially on a normal subgroup
- Revisiting a theorem of Higman
- Cohomological finiteness conditions
Summary
Introduction
Control of fusion
Let G be a finite group, and let p be a prime number.
Definition 1.1. We say that a subgroup H of G controls the fusion of p-subgroups of G if the following two conditions are fulfilled:
(C1) H contains a Sylow p-subgroup Sp of G,
(C2) whenever P is a subgroup of Sp and g is an element of G such that gPg-1 ⊆ Sp, there exist z in the centralizer CG(P) of P in G, and h in H, such that g = hz.
Example 1.2. (The basic example) We denote by Op′(G) the largest normal subgroup of G with order prime to p. Then if H is a subgroup of G which “covers the quotient” G/OP′(G) (i.e., if G = HOP′(G)), then H controls the fusion of p-subgroups of G.
The following two results provide fundamental examples where the converse is true. The first one is due to Frobenius and was proved in 1905. The second one was proved by Glauberman for the case p = 2 (see [Gl]), and for odd p it is a consequence of the classification of non abelian finite simple groups (see also [Ro] for an approach not using the classification).
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- Groups '93 Galway/St Andrews , pp. 58 - 79Publisher: Cambridge University PressPrint publication year: 1995
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