Special Linear Group is Normal Subgroup of General Linear Group

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 a normal subgroup of the general linear group $\GL {n, K}$.

Proof

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

It remains to be shown that $\SL {n, K}$ is a normal subgroup of $\GL {n, K}$.

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): general linear group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): general linear group