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, let me make the following semi-conjecture, which, if true, would make me very happy:



> 
> The irreducible complex representations of Sylow 2-subgroups of $\rm{PGL}\_n(\mathbb{F}\_q)$ 
>  have trivial Schur index.
> 
> 
> 


Is this known? 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.



Sylow subgroups of projective general linear groups
The irreducibles of the Sylow 2-subgroups of ${\rm GL}(n,q)$, $q$ odd, have indeed trivial Schur indices: Let $S$ be such a Sylow 2-subgroup. First, observe that the natural module $(\mathbb{F}\_q)^n$ splits into a sum of simple modules $U\_1\oplus \dotsb \oplus U\_l$, and the dimension of each simple module is a power of $2$. This shows that $S \cong S\_1 \times \dotsb \times S\_l $, where the $S\_i$'s are Sylow 2-subgroups of a ${\rm GL}(2^k, q)$. Thus, w.l.o.g. we may assume that $n=2^k$. Then I use induction on $k$. First note that if $S$ is a Sylow 2-subgroup of ${\rm GL}(2^k, q)$, then 
$$ T=\lbrace \begin{pmatrix} s & \\ & t \end{pmatrix} \mid s, t\in S \rbrace 
 \cup \lbrace \begin{pmatrix} & s \\ t & \end{pmatrix} \mid s,t\in S\rbrace
 \cong S\wr C\_2 $$
is a Sylow 2-subgroup of ${\rm GL}(2^{k+1}, q)$, except when $k=0$ and $q\equiv 3\mod 4$. This follows from the description in Derek Holt's answer, but it can also be seen directly by observing that $T$ has the right order.  

Write $N=S\times S$, so that $T= C\_2 N$, and let $\chi\in {\rm Irr} (T)$. Three cases have to be considered:  

1. We have $\chi\_N \in {\rm Irr} (N)$. By induction, $\chi\_N$ has trivial Schur index. By Lemma~10.4 in Isaacs' character theory book, for example, it follows that $\chi$ has trivial Schur index.  

2. $\chi\_N = \theta + \theta^g$, where $\theta$ and $\theta^g$ are not Galois conjugate. 
Then $\chi = \theta^T$ and $\theta$ have the same field of values and the same Schur index (again, see Lemma~10.4 in Isaacs).  

3. $\chi\_N = \theta + \theta^g$, where $\theta$ and $\theta^g$ are Galois conjugate. This means that $\chi=\theta^T$, but $|\mathbb{Q}(\theta):\mathbb{Q}(\chi)|=2$. Now a representation $R\colon N\to {\rm GL}(\chi(1), \mathbb{Q}(\chi) )$ affording the character $\theta+\theta^g$ can be extended to a representation over the same field, since $N$ has a complement (of order 2) in $T$.  

To get the induction going when $q\equiv 3\mod 4$, one has to check that the Sylow 2-subgroup of ${\rm GL}(2,q)$ has trivial Schur indices, but that is clear since it's a semidihedral group.


