Lower central series of general linear group stabilizes at special linear group

Statement

Let $k$ be a field and $n$ be a natural number. Then, the lower central series of $Gl_{n}(k)$ stabilizes in one step at $Sl_{n}(k)$. In other words, it looks like:

$GL_{n}(k)\geq SL_{n}(k)\geq SL_{n}(k)\geq SL_{n}(k)\geq \dots$.

This holds under either of these conditions:

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

Note that for $n\geq 3$ and $k$ having exactly two elements, $GL_{n}(k)=SL_{n}(k)$, so the stabilization occurs at the first step itself. Also, for $n=1$, the result holds trivially, since $GL_{1}(k)$ is Abelian and $SL_{1}(k)$ is trivial.

Related facts

• Special linear group is perfect: This holds under slightly more restrictive circumstances: $k$ should have more than three elements.
• Commutator subgroup of general linear group is special linear group: This is a weaker version, and holds under the same circumstances.

Facts used

1. Every elementary matrix is the commutator of an invertible and an elementary matrix: This holds under the same hypotheses: $n\geq 3$ or $k$ has at least three elements.
2. Elementary matrices generate the special linear group

Proof

The commutator subgroup is the special linear group

By facts (1) and (2), the commutator subgroup of $GL_{n}(k)$ contains $SL_{n}(k)$. On the other hand, $Sl_{n}(k)$ is the kernel of a homomorphism from $Gl_{n}(k)$ to the multiplicative group of $k$, which is Abelian. Thus, the commutator subgroup of $Gl_{n}(k)$ is contained in $Sl_{n}(k)$. Combining these facts, we get that the commutator subgroup of $Gl_{n}(k)$ equals $SL_{n}(k)$.

Further members of the lower central series

By fact (1), every elementary matrix can be expressed as the commutator between an element of $GL_{n}(k)$ and an element of $SL_{n}(k)$. Fact (2) thus yields:

$SL_{n}(k)\leq [GL_{n}(k),SL_{n}(k)]$.

On the other hand, we know that the right side is contained in $SL_{n}(k)$, since $SL_{n}(k)=[GL_{n}(k),GL_{n}(k)]$. Thus, we get:

$SL_{n}(k)=[GL_{n}(k),GL_{n}(k)]$.

Thus, the lower central series stabilizes at $SL_{n}(k)$.