$SL(2,q)$ when $q$ is odd

Question

I need some information about $SL(2,q)$, when $q$ is odd. Thank you in advance.

Answer 1

Let $G = {\rm SL}(2,q)$. When $q$ is odd $Z := Z(G)$ has order $2$, and $Z$ contains the unique element of order $2$ in $G$. You already have a list of the subgroups of $G/Z = {\rm PSL}(2,q)$, and you can use that to find the subgroups of $G$.

Let $H$ be a subgroup of $G$. Then $\bar{H} = HZ/Z$ is on your list of subgroups.

If $|\bar{H}|$ is odd, then $HZ \cong \bar{H} \times Z$, and then either $Z \le H$ and $H = HZ$, or $H \cong \bar{H}$ and $H$ is the unique complement of $Z$ in $HZ$.

If $|\bar{H}|$ is even, then $|H|$ is even, and so we must have $Z < H$, because $Z$ contains the unique element of order $2$ in $G$. So in this case $H$ is uniquely determined as the complete inverse image of $\bar{H}$ in $G$.