Aschbacher's description of Sylow 2-subgroups

Question

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?

• Aschbacher, Michael. “A characterization of Chevalley groups over fields of odd order.” Ann. of Math. (2) 106 (1977), no. 2, 353–398. MR498828
• Aschbacher, Michael. “A characterization of Chevalley groups over fields of odd order. II.” Ann. of Math. (2) 106 (1977), no. 3, 399–468. MR498829
• Aschbacher, Michael. “On finite groups of Lie type and odd characteristic.” J. Algebra 66 (1980), no. 2, 400–424. MR593602
• Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald. The classification of the finite simple groups. Number 3. Part I. Chapter A. Almost simple K-groups. Mathematical Surveys and Monographs, 40.3. American Mathematical Society, Providence, RI, 1998. xvi+419 pp. ISBN: 0-8218-0391-3 MR1490581
• Harada, Koichiro; Lang, Mong Lung. “Indecomposable Sylow 2-subgroups of simple groups.” Acta Appl. Math. 85 (2005), no. 1-3, 161–194. MR2128910

Answer 1

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.