Special linear group is quasisimple
Let be a field and a natural number greater than . The special linear group is a quasisimple group: it is a perfect group and its inner automorphism group is a simple group, except in the following case: and has two or three elements.
In other words, if or has at least four elements, the special linear group is quasisimple.
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.