Memoirs of the American Mathematical Society 1998; 123 pp; softcover Volume: 136 ISBN10: 0821808702 ISBN13: 9780821808702 List Price: US$50 Individual Members: US$30 Institutional Members: US$40 Order Code: MEMO/136/649
 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) \(G = \langle P_ii \in I\rangle\), (2) \(B\) is the normalizer of a \(pSylow\)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 Lietype 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 Titsgeometries and the classification of quasithin groups. Readership Graduate students and research mathematicians working in classical linear algebraic groups. Table of Contents  Introduction
 Weak \((B,N,)\)pairs of Rank 2
 Modules
 The Graph \(\Gamma\)
 The structure of \(\overline L_\delta\) and \(\overline Z_\delta\)
 The case \(b\geq 2\)
 The case \(b=0\)
 The case \(b=1\) and the proof of Theorems 1 and 4
 The proof of Theorems 2 and 3
