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I'm looking for a few more examples of using Aschbacher (1980)'s “fundamental SL2 subgroups” description of Sylow 2-subgroups of finite groups of Lie type in odd characteristic. I do not yet understand his original paper, nor the (brief) discussion in GLS (1998), but the use in Harada–Lang (2005) was very compelling.

Can anyone recommend other papers or presentations that might make the description more clear?

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  • $\begingroup$ Jack, I don't quite understand your question. Do you want examples of Aschbacher's description being used to prove interesting results, or do you want a source that gives a clearer treatment of Aschbacher's result (or both)? $\endgroup$ – Nick Gill May 13 '14 at 17:04
  • $\begingroup$ Both are fine. I basically want an overview, an intuitive explanation, with an example of it being used. Aschbacher's treatment is likely clear, but it is long (150 pages). Harada–Lang is quite short (a page or so is all that is needed), so shows the power of learning one or two results from Aschbacher. However, it doesn't quite help me learn those results. I believe this technique is intended to be sort of a default way to understand groups in odd characteristic, so assume that more motivational presentations have been given. $\endgroup$ – Jack Schmidt May 13 '14 at 17:23
  • $\begingroup$ OK, that's clear enough. This might be relevant: ams.org/mathscinet-getitem?mr=842439 (I can't get hold of a copy so can't be sure.) $\endgroup$ – Nick Gill May 13 '14 at 18:25
  • $\begingroup$ That was actually very helpful in a simple way: this description is called the "classical involution theorem" which I've studied before but did not recognize. en.wikipedia.org/wiki/Classical_involution_theorem $\endgroup$ – Jack Schmidt May 13 '14 at 18:31
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Perhaps this question should be a Community Wiki, because there are potentially many applications that could be listed. My favourite is, perhaps, Bill Kantor's use of Aschbacher's result to classify the primitive permutation groups of odd degree:

Kantor, William M. Primitive permutation groups of odd degree, and an application to finite projective planes. J. Algebra 106 (1987), no. 1, 15–45.

Once this classification is complete, Kantor applies the result to the study of finite projective planes admitting a point-primitive automorphism group. He proves that either the plane is Desarguesian, or the group is a regular Frobenius group.

In addition to giving an example of an application of Aschbacher's result, the paper includes a short section entitled NOTES ON ASCHBACHER'S CLASSICAL INVOLUTION THEOREM which may help to describe that result a little more.

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  • $\begingroup$ Thanks again! Sections 1 and 3 are mostly about the direct consequences of the CIT. $\endgroup$ – Jack Schmidt May 13 '14 at 20:01
  • $\begingroup$ Pleasure. Glad to be useful! $\endgroup$ – Nick Gill May 13 '14 at 20:08

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