# Rank 3 Amalgams

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*Bernd Stellmacher; Franz Georg Timmesfeld*

Let \(G\) be a group, \(p\) a fixed prime,
\(I = {1,...,n}\) and let \(B\) and \(P_i, i \in
I\) be a collection of finite subgroups of \(G\). Then
\(G\) satisfies \(P_n\) (with respect to \(p\),
\(B\) and \(P_i, i \in I\)) if:

(1) \(G = \langle P_i|i \in I\rangle\),

(2) \(B\) is the normalizer of a \(p-Sylow\)-subgroup in \(P_i\),

(3) No nontrivial normal subgroup of \(B\) is normal in \(G\),

(4) \(O^{p^\prime}(P_i/O_p(P_i))\) is a rank 1 Lie-type group in char \(p\) (also including solvable cases).

If \(n = 2\), then the structure of \(P_1, P_2\) was
determined by Delgado and Stellmacher. In this book the authors treat
the case \(n = 3\). This has applications for locally finite,
chamber transitive Tits-geometries and the classification of quasithin
groups.

#### Table of Contents

# Table of Contents

## Rank 3 Amalgams

- Contents vii8 free
- 0 Introduction 110 free
- 1 Weak (B,N)-pairs of Rank 2 1524 free
- 2 Modules 2231
- 3 The Graph Γ 2736
- 4 The Structure of L[sub(δ)] and Z[sub(δ)] 3847
- 5 The Case b ≥ 2 5362
- 6 The Case b = 0 6473
- 7 The Case b = 1 and the Proof of Theorems 1 and 4 7584
- 8 The Proof of Theorems 2 and 3 92101