# How can I explain the sketch of the proof of classification of finite simple groups to someone who knows only basics of algebra?

This question was popped up in my mind when I read Finnish Wikipedia. How can I explain the sketch of the proof to layman? Is it worth to explain for example Ree groups in the text or just say something general like that kind of groups are important without going to details?

• What are "the basics of Algebra"? – Mark Bennet Sep 14 '13 at 20:59
• Lets say to someone who knows simple groups, Lagrange's theorem and Sylow theorem. – FinnWiki Sep 15 '13 at 6:40
• Do you just want to sketch the result or indeed the proof? – j.p. Sep 15 '13 at 9:58
• I would like to have a sketch of the proof. – FinnWiki Sep 15 '13 at 10:07
• You can take a look either at www.ams.org/notices/199502/solomon.pdf or at the first volume of "The Classification of the Finite Simple Groups" by Daniel Gorenstein, Richard Lyons and Ronald Solomon (which - if I remember correctly - contains a sketch of the proof). For a sketch of the proof you might not need to explain how to construct the families of groups of Lie type, as in the proof all work is "done in characteristic 2". – j.p. Sep 15 '13 at 12:49

## 1 Answer

An outline of the proof of CSFG (Classification of finite simple groups) is given in the "Surveys of the AMS, Volume 172, by Aschbacher, Lyons, Smith, and Solomon. From the summary:

Since “most” finite simple groups $G$ are in fact matrix groups over finite fields, an early result determining the overall shape of the CFSG was the Dichotomy Theorem---which shows that an abstract simple group $G$G (away from cases with 2-subgroups of rank at most 2) is either:
of COMPONENT TYPE (resembling a group over a field of odd order), or of
CHARACTERISTIC 2-TYPE (resem- bling a group in characteristic 2).
The treatment of the “odd case”, namely component type, was based on Aschbacher’s notion of a quasisimple component L of STANDARD FORM, in the centralizer in G of an element t of order 2. The various possible L were treated by Aschbacher and various other researchers. The treatment of the remaining “even case”, namely characteristic 2-type, was obtained via suitable analogies of the above case divisions---but replacing t by an element of ODD prime order p. Here the initial “small” case corresponds to QUASITHIN groups G; namely where the rank of suit- able p-subgroups is at most 2. This situation involves many complications; it was eventually treated in a lengthy work of Asch- bacher and Smith. For the remaining cases involving p-subgroups of rank at least 3, the above Dichotomy is replaced by a Trichotomy---established by Gorenstein and Lyons (with contributions from Asch- bacher). The three branches which emerge are: a (p-component) branch called Standard Type; a (roughly characteristic p-type) branch leading to “GF(2) type”; and a further “disconnected” branch called the Uniqueness Case. The groups of standard type were determined by Gilman and Griess. The groups of GF(2) type were determined by various authors, including Aschbacher, Timmesfeld, and Smith. And the final contradiction in the CFSG (although quasithin groups were chronologically the last to be treated) was established by Aschbacher, who showed that no group can actually satisfy the Uniqueness Case.