With three preliminary volumes under our belt, we at last turn directly to
the proof of the Classification Theorem; that is, we begin the analysis of a minimal
counterexample G to the theorem. Thus G is a DC-proper simple group. Specifically,
the heart of this volume provides a collection of uniqueness and pre-uniqueness
theorems for G, proved in Chapters 2 and 3.
Chapter 2 is primarily dedicated to the proof of a fundamental 2-uniqueness
theorem of Michael Aschbacher, which we call Theorem ZD. Historically there is a
sequence of basic papers leading up to Theorem ZD. First is a fundamental paper of
Michio Suzuki [Su4] on 2-transitive permutation groups; the result of that paper is
one of our Background Results [PI, Part II]. Building on that, Helmut Bender [Be3]
established the Strongly Embedded Subgroup Theorem, which we call Theorem SE.
Finally Aschbacher [A3] proved Theorem ZD and its corollaries, based on Bender's
theorem. In both Bender's and Aschbacher's papers, the first major step is to
establish that the group under investigation acts 2-transitively on a certain set.
Aschbacher's arguments for this step are modelled closely on Bender's and so it
seems natural to combine these proofs. This unified proof constitutes Sections 3-7
of Chapter 2, and follows the original arguments of Bender and Aschbacher closely.
However, our treatment of Section 7 benefits considerably from unpublished notes of
David Goldschmidt. After this point the proofs by Bender and Aschbacher diverge
both in the originals and here. Indeed we follow the originals quite closely at most
points, and we have made use of Peterfalvi's revision of this proof [PI] in Sections
8-11 of Chapter 2.
The remainder of Chapter 2 treats three corollaries of Theorem ZD. The clas-
sification of DC-proper simple groups with a 2-uniqueness subgroup, Theorem SU,
combines a proof of Aschbacher's theorem [A3] on groups with a proper 2-generated
core, but only for DC-proper simple groups, with ideas of Koichiro Harada [HI]. Fi-
nally the important Theorems SA and SF of Goldschmidt [Go5] and Holt [Hoi] are
proved only for DC-proper simple groups of even type, which circumvents many of
the difficulties in the original papers.
Our approach to Theorem ZD and the consequent Theorems SA and SZ in
Chapter 2 is to formulate them not just for G but for an arbitrary finite group X.
Thus for the major part of this chapter, we depart from our original plan regarding
2-uniqueness theorems, as announced in [Ii] and carried out in preprints preliminary
to this chapter. That strategy, which remains workable, was to prove the results
only for the DC-proper simple group G; in the case of Theorem SE, for example,
making central use of a classification of DC-groups M whose set of involutions is
permuted transitively by some subgroup of M of odd order. Indeed, such a group
M can be proved either to be solvable or to have a unique composition factor of
even order, which is isomorphic to Z^tf) for some q = 3 mod 4. We have returned