# 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

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: