Special linear group is perfect

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

Statement

Let $k$ be any field. Consider the special linear group $SL_{n}(k)$: the group of $n\times n$ matrices over $k$ that have determinant $1$. Then, the following are true:

• For $n\geq 3$, $SL_{n}(k)$ is a perfect group for any field $k$.
• For $n=2$, $SL_{n}(k)$ is a perfect group if $k$ has more than three elements.

Related facts

• Derived subgroup of general linear group is special linear group
• Symplectic group is perfect
• Derived subgroup of orthogonal group is special orthogonal group

Facts used

1. Elementary matrices generate the special linear group
2. Every elementary matrix is a commutator of unimodular matrices: If $n\geq 3$ or $n=2$ and $k$ has more than three elements, every elementary $n\times n$ matrix is a commutator of two elements of $SL_{n}(k)$.

Proof

The proof follows by piecing together facts (1) and (2).