Rank 3 Amalgams (Memoirs of the American Mathematical Society)

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.