# Sylow subgroup of a subgroup 5

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.

• The answer to your question is yes because ${\rm SL}(n,K)$ is a normal subgroup of ${\rm GL}(n,K)$ for all fields $K$. – Derek Holt Jun 23 at 6:53

If $$H \leq G$$, then a $$p$$-subgroup of $$H$$ is also a $$p$$-subgroup of $$G$$, and so is contained in a Sylow $$p$$-subgroup of $$G$$, and every $$p$$-subgroup is contained in a Sylow $$p$$-subgroup. In other words, yes, a Sylow $$p$$-subgroup of $$H$$ must be contained in a Sylow $$p$$-subgroup of $$G$$.
In the case that $$H \trianglelefteq G$$, then since all Sylow $$p$$-subgroups of any group are conjugate (for a fixed $$p$$), and $$H^{g} = H$$ (conjugation of $$H$$ by $$g$$) for all $$g \in G$$, then we also have conversely that every Sylow $$p$$-subgroup of $$G$$ meets $$H$$ in a Sylow $$p$$-subgroup.