Sylow subgroups of $\text{SL}_2(q)$.

Question

Let $p,q$ be primes such that $p$ is a divisor of $|\text{SL}_2(q)|=(q-1)q(q+1)$. Hence $\text{SL}_2(q)$ admits non-trivial Sylow subgroups. I am interested in the isomorphism type of the $p$-Sylow. From what I understand, if $p<q$ are odd primes, then a $p$-Sylow subgroup of $\text{SL}_2(q)$ is cyclic.

However, in the case $p=2$ or $p=q$, I am not sure.

I tried to use Suzuki paper(http://www.jstor.org/stable/2372591?seq=1#page_scan_tab_contents), but it is rather complicated and I hoped there is an easier approach.

Thanks in advance for any help.

Answer 1

We know that the group $SL_2(3)$ has $Q_8$ as $2$-Sylow-subgroup, which certainly is not a cyclic group - see here. This is the case $p=2$, $q=3$, satisfying $p\mid (q-1)q(q+1)$.

Edit: For odd primes $p$, every Sylow $p$-subgroup of $SL(2,q)$ is indeed cyclic (see here, G6f3 c)).

Answer 2

${\rm SL}(2,q)$ has a unique element of order $2$, namely $-I_2$. So the same applies to its Sylow $2$-subgroups. There is theorem that the only $2$-groups with this property are cyclic and generalized quaternion. Since the Sylow $2$-subgroups are not cyclic, they are geenralized quaternion.