# Special Linear Group is Normal Subgroup of General Linear Group

Jump to navigation
Jump to search

## 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 $\GL {n, K}$.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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**