What is known about the Sylow 2-subgroups of $\rm{PGL}_n(\mathbb{F}_q)$, where $q$ is any prime power? For example, according to Theorem 7.9 in Isaacs's Character Theory, these Sylow 2-subgroups cannot be generalised quaternion. Is there a classification of all these Sylows available?
Concretely, I let me make the following semi-conjecture, which, if true, would be make me very happyto hear something along the lines of "the :
The irreducible complex representations of Sylow 2-subgroups of $\rm{PGL}_n(\mathbb{F}_q)$ have trivial Schur index"index.
Is this trueknown? Any references to literature that discusses such questions would be very welcome.
If anyone has information on Sylow $l$-subgroups for arbitrary $l$, I would also be very interested.
