I am aiming for looking for all the Sylow subgroups of classical groups, which gives me a seemingly elementary question: what can we say about Sylow subgroups of a subgroup if we know all Sylow subgroup of the original one?
In my case, I know some Sylow subgroups of $GL(n,F_q)$ in which $F_q$ is a finite field and I wonder can I find some Sylow subgroups of $SL(n,F_q)$ from those and vice versa.
I found this post Counterexample of Sylow subgroups of a subgroup and if the writer is correct, then let $P$ be a Sylow subgroup of $GL$, then $P\cap SL$ is a Sylow subgroup of $SL$ since $SL$ is a normal subgroup of $GL$.
This is my question, given a Sylow subgroup $S$ of $SL$, then is there a Sylow subgroup $P$ of $GL$ s.t. $S=P\cap SL$?
I looked for examples but failed. However, if this is true, then there would be a very beautiful relation between Sylow subgroups of $SL$ and $GL$, which is too good to be true.