**Memoirs of the American Mathematical Society**

2019;
182 pp;
Softcover

MSC: Primary 20;
Secondary 55

Print ISBN: 978-1-4704-3520-2

Product Code: MEMO/257/1236

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

**Electronic ISBN: 978-1-4704-4955-1
Product Code: MEMO/257/1236.E**

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

# On Fusion Systems of Component Type

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*Michael Aschbacher*

This memoir begins a program to classify a large subclass of the
class of simple saturated 2-fusion systems of component type. Such a
classification would be of great interest in its own right, but in
addition it should lead to a significant simplification of the proof of
the theorem classifying the finite simple groups.

Why should such a
simplification be possible? Part of the answer lies in the fact that
there are advantages to be gained by working with fusion systems rather
than groups. In particular one can hope to avoid a proof of the
B-Conjecture, a important but difficult result in finite group theory,
established only with great effort.

#### Table of Contents

# Table of Contents

## On Fusion Systems of Component Type

- Cover Cover11
- Title page i2
- Introduction 18
- 0.1. Fusion systems 18
- 0.2. Local group theory 29
- 0.3. The local theory of fusion systems 310
- 0.4. The program 310
- 0.5. From fusion systems to groups 512
- 0.6. The main theorems 613
- 0.7. Tight embedding 613
- 0.8. Pumpups 815
- 0.9. Component theorems for fusion systems 815
- 0.10. Tame systems 916
- 0.11. J-components 916
- 0.12. Standard subsystems 1017
- 0.13. Tight split extensions 1118
- 0.14. Quaternion fusion packets and Walter’s Theorem 1219
- 0.15. Normalizers and centralizers of components 1320

- Chapter 1. Preliminaries 1522
- Chapter 2. Some Lemmas on Fusion Systems 2734
- 2.1. The normalizer of a component 2734
- 2.2. Some properties of the normalizer of a component 3138
- 2.3. Some lemmas on fusion systems 3239
- 2.4. Essential subgroups in fusion systems. 3441
- 2.5. Automorphisms of quasisimple systems 3542
- 2.6. A lemma from the component paper 4148
- 2.7. Random lemmas on fusion systems 4653
- 2.8. The kernel of ℱ on subgroups of 𝒪_{𝓅}(ℱ) 4855

- Chapter 3. Tight embedding 5158
- 3.1. Tightly embedded subsystems 5360
- 3.2. TI-subgroups of 2-groups 5764
- 3.3. Tight embedding when 𝑝=2 6168
- 3.4. Oversystems of tightly embedding subsystems 6269
- 3.5. The system 𝒪 6370
- 3.6. 𝒯=∅ 6572
- 3.7. The proof of Theorem 1 7784
- 3.8. |𝑇|>2 for 𝑇∈𝒯 7885
- 3.9. 𝑇∈𝒯 with Φ(𝒯)̸=1 8188
- 3.10. The proof of Theorems 3.0.2 and 3.0.3 8693
- 3.11. The proof of Theorem 3.0.4 8794

- Chapter 4. More on tight embedding 93100
- Chapter 5. Split extensions 111118
- Chapter 6. Component combinatorics 117124
- Chapter 7. The proof of Theorem 2 141148
- Chapter 8. Terminal components 165172
- Chapter 9. Standard subsystems 169176
- Bibliography 181188
- Back Cover Back Cover1194