Projective special linear group is simple

This article gives the statement, and possibly proof, of a particular group or type of group (namely, Projective special linear group (?)) satisfying a particular group property (namely, Simple group (?)).

Statement

Let $k$ be a field and $n$ be a natural number greater than $1$. Then, the projective special linear group $PSL_{n}(k)$ is a simple group provided one of these conditions holds:

• $n\geq 3$.
• $k$ has at least four elements.

Facts used

1. Special linear group is quasisimple

Related facts

• Special linear group is perfect
• Special linear group is quasisimple
• Projective special linear group equals alternating group in only finitely many cases

Proof

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