Amalgam methods applied to the structure of the finite simple groups of characteristic 2 type

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The purpose of this project is to classify the finite simple groups satisfying a certain condition (A). The actual work is stated in the following three-step procedure. 1. Classify the isomorphism classes of the amalgams (X, S, Y) of a finite 2-group S and finite groups X, Y containing S as their common Sylow 2-subgroup satisfying the conditions : (1) both O_2(X) and O_2(Y) are nonidentity ; (2) O_2(X, Y) is identity : (3) an additional condition may be fulfilled. 2. For a finite simple group G satisfying the condition (A), find an amalgam (X, S, Y) of subgroups S, X, Y of G satisfying the above three conditions. 3. Classify the finite simple groups G satisfying the condition (A) by judging which isomorphism class in the first step the amalgam of G found in the second step is belonging to.Basically, our result is divided into three categories. First of all, when the condition (A) is said to be 'simple groups of characteristic 2 type all of whose 2-local subgroups are solvable,' we study the third step above, and get a new proof of the classification of such simple groups. Our proof is much shorter and conceptually easier than the existing one. Next, more generally, when the condition (A) is 'simple groups of characteristic 2 type,' we study the third step above, and determine the structure of quadratic module of finite simple groups. In the analysis of 2-local subgroups of finite simple groups of characteristic 2 type, we often encounter modules over GF(2) having quadratic 2-subgroups. So we can expect that this result will apply to the study of the third step above. Finally, when the condition (A) is 'simple groups of characteristic 2 type,' we also study the geometric structure of the prime graphs of finite simple groups. Though the importance of the connected components of only odd primes was already well known, we focus attention on its connected components containing the prime 2 and even local subgroups instead of 2-local subgroups.