In "Abelian l-Adic Representations and Elliptic Curves", page 169 (link provided at the end), Serre proposes the following lemma:
"Let $X$ be a closed subgroup of $\text{SL}_{2}(\mathbb{Z}_{p})$ whose image in $\text{SL}_{2}(\mathbb{F}_{p})$ is $\text{SL}_{2}(\mathbb{F}_{p})$. Assume $p \geq 5$. Then $X=\text{SL}_{2}(\mathbb{Z}_{p})$."
I'm having a lot of trouble understanding what this lemma is actually telling me about the group structure of, say $\text{SL}_{2}(\mathbb{Z}/{p}^{k}\mathbb{Z}).$ Any help understanding this lemma would be greatly appreciated. As well as this, the excerises that follow ask
- a) Generalise this lemma for $\text{SL}_{d}(\mathbb{Z}_{p})$. This, as far as i can see, is identical to the proof of the $d=2$ case.
b) Show that the only closed subgroup of $\text{SL}_{d}(\mathbb{Z}_{3})$ which maps onto $\text{SL}_{d}(\mathbb{Z}/3^{2}\mathbb{Z})$ is $\text{SL}_{d}(\mathbb{Z}_{3})$ itself.
c) Show that the only closed subgroup of $\text{SL}_{d}(\mathbb{Z}_{2})$ which maps onto $\text{SL}_{d}(\mathbb{Z}/2^{3}\mathbb{Z})$ is $\text{SL}_{d}(\mathbb{Z}_{2})$ itself.
These last two im having a really hard time grasping, although i think its because the resitiction $p \geq 5$ can be extended to $p^{k} \geq 5$, which $3^{2}$ and $2^{3}$ are, but im not sure.
https://www.dpmms.cam.ac.uk/~ty245/camonly/SerreBook.pdf
(lemma in question is Lemma 3 on page 169, and excerises are on page 173)