What do Sylow 2-subgroups of finite simple groups look like? What do Sylow 2-subgroups of finite simple groups look like?
It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision many answers. I tried to give an answer of "broad strokes with references." It would be reasonable to answer just for a single infinite family of finite simple groups too. Here are some specific aspects that could be addressed.


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*Which finite simple groups have the same Sylow 2-subgroup?

*Are there ways in which Sylow 2-subgroups of finite simple groups are vastly different from a "random" or "typical" 2-group? (For instance, are the nilpotency class and derived length related in some special way compared to the general 2-group.)

*What do the normalizers of the Sylow 2-subgroups look like (in cases where we understand the Sylow itself well)?

*What are its characteristic, normal in the normalizer, or normal subgroups?

*What are the possible fusion systems?

 A: Normalizers and top fusion
Kondratʹev (2005) contains a very clear description of the normalizers of Sylow 2-subgroups $S$ of finite simple groups. Typically there is not much going on: $N_G(S) = S$ unless:


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*$G$ is a group of Lie type in even characteristic (then $N_G(S)$ is a Borel subgroup and $N_G(S)/S$ is a maximally-split maximal torus, which is usually not the identity unless $q=2$)

*$S$ is abelian, so described in Walter's theorem (Janko-Ree or certain PSL2)

*$N_G(S)=S C_G(S)$ as in PSL, PSU, E6 and 2E6

*$N_G(S)/S$ is an elementary abelian 3-group ($J_2,J_3, HN,PSp_{2m}$ for most values of $m$)


So other than groups of even characteristic, the automorphisms of $S$ induced from $G$ are very limited, usually inner, but with an order 7 in the Janko–Ree case and order 3 in the Janko–Ree, three more sporadics, and most of the symplectic groups.
Bibliography


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*Kondratʹev, A. S.
“Normalizers of Sylow 2-subgroups in finite simple groups.” (original)
Mat. Zametki 78 (2005), no. 3, 368--376; translation in 
Math. Notes 78 (2005), no. 3-4, 338–346
MR2227510
