A reference request for $SL(2,q)$ being quasisimple for prime powers $q\ge 4$.

Question

Note: This is a reference-request question and thus does not need the usual type of context.

The Question:

What is a reference for $SL(2,q)$ being quasisimple for prime powers $q\ge 4$?

Background:

I know from various sources, such as my PhD supervisor and this GroupProps page, that the claim is true; I can't seem to find a good reference for it though. I need it for a review I'm writing.

It's not in any of the obvious places. My search terms have been along the lines of:

• SL(2,q) "quasisimple",
• special linear group over finite field quasisimple, and
• reference SL(2,q) quasisimple.

An Idea:

The proof is a combination of knowing that:

1. $SL(2,q)$ is perfect and
2. $PSL(2,q)$ is simple

for the values of $q$ above. References for those would suffice.

Answer 1

Here are two possibilities.

1. In Huppert's German book "Endliche Gruppen I", it is proved in Satz (= Theorem) 6.10 on page 181 that ${\rm SL}(n,K)$ is perfect for any field $K$ and $n \ge 2$, except when $n=2$ and $|K| =2$ or $3$, and then in Satz 6.13 on page 182 that ${\rm PSL}(n,K)$ is simple for the same $n$ and $K$.

2. (Much easier to read, and almost what you want) In Rotman's book "An Introduction to the Theory of Groups", Fourth Edition, Theorem 8.13, page 225, it is proved that ${\rm PSL}(2,q)$ is simple for $q \ge 4$, but what is actually proved is that, for $q \ge 4$, any normal subgroup of ${\rm SL}(2,q)$ that contains a non-scalar matrix is equal to ${\rm SL}(2,q)$. So to complete the quasisimplicity proof we just need to prove that the derived group of ${\rm SL}(2,q)$ contains a nonscalar matrix or, equivalently, that ${\rm PSL}(2,q)$ is nonabelian, which is not very hard!

PS: Another reference is I.M. Isaacs "Finite Group Theory", Thm 8.32 ${\rm SL}(n,q)$ is perfect and Thm 8.33 ${\rm PSL}(n,q)$ is simple, both for $n \ge 2$, $(n,q) \ne (2,2), (2,3)$.