General Linear Group is not Abelian/Proof 1

Theorem

Let $K$ be a field whose zero is $0_K$ and unity is $1_K$.

Let $\GL {n, K}$ be the general linear group of order $n$ over $K$.

Then $\GL {n, K}$ is not an abelian group.

Proof

From Special Linear Group is Subgroup of General Linear Group we have that the special linear group $\SL {n, K}$ is a subgroup of $\GL {n, K}$.

From Special Linear Group is not Abelian, $\SL {n, K}$ is not abelian.

From Subgroup of Abelian Group is Abelian it follows by the Rule of Transposition that $\GL {n, K}$ is not abelian.

$\blacksquare$