Special Linear Group is not Abelian

Theorem

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

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

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

Proof

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

From Matrix Multiplication is not Commutative‎ it follows that $\SL {n, K}$ is not abelian.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 29 \alpha \ (5)$
• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP