The present book, “The Classification of Finite Simple Groups: Groups of
Characteristic 2 Type”, completes a project of giving an outline of the proof of
the Classification of the Finite Simple Groups (CFSG). The project was begun by
Daniel Gorenstein in 1983 with his book [Gor83]—which he subtitled “Volume 1:
Groups of Noncharacteristic 2 Type”. Thus we regard our present discussion of
groups of characteristic 2 type as “Volume 2” of that project.
The Classification of the Finite Simple Groups (CFSG) is one of the premier
achievements of twentieth century mathematics. The result has a history which, in
some sense, goes back to the beginnings of proto-group theory in the late eighteenth
century. Many classic problems with a long history are important more for the
mathematics they inspire and generate, than because of interesting consequences.
This is not true of the Classification, which is an extremely useful result, making
possible many modern successes of finite group theory, which have in turn been
applied to solve numerous problems in many areas of mathematics.
A theorem of this beauty and consequence deserves and demands a proof ac-
cessible to any mathematician with enough background in finite group theory to
read the proof. Unfortunately the proof of the Classification is very long and
complicated, consisting of thousands of pages, written by hundreds of mathemati-
cians in hundreds of articles published over a period of decades. The only way
to make such a proof truly accessible is, with hindsight, to reorganize and rework
the mathematics, collect it all in one place, and make the treatment self-contained,
except for some carefully written and selected basic references. Such an effort is
in progress in the work of Gorenstein, Lyons, and Solomon (GLS) in their series
beginning with [GLS94], which seeks to produce a second-generation proof of the
However in the meantime, there should at least be a detailed outline of the
existing proof, that gives a global picture of the mathematics involved, and explicitly
lists the papers which make up the proof. Even after a second-generation proof is
in place, such an outline would have great historical value, and would also provide
those group theorists who seek to further simplify the proof with the opportunity
to understand the approach and ideas that appear in the proof. That is the goal of
this volume: to provide an overview and reader’s guide to the huge literature which
makes up the original proof of the Classification.
Soon after the apparent completion of the Classification in the early 1980s,
Daniel Gorenstein began a project aimed at giving an outline of the original proof.
He provided background in a substantial Introduction [Gor82], in particular dis-
cussing the partition of simple groups into groups of odd characteristic and groups
of characteristic 2 type. Then in Volume 1 [Gor83] he described the treatment of