Special linear group is quasisimple

Statement

Let $k$ be a field and $n$ a natural number greater than $1$. The special linear group $SL_{n}(k)$ is a quasisimple group: it is a perfect group and its inner automorphism group is a simple group, except in the following case: $n=2$ and $k$ has two or three elements.

In other words, if $n\geq 3$ or $k$ has at least four elements, the special linear group $SL_{n}(k)$ is quasisimple.

Facts used

1. Special linear group is perfect
2. Projective special linear group is simple

Proof

The proof follows from facts (1) and (2), and the fact that the inner automorphism group of the special linear group is the projective special linear group.