## Special Linear Group is a Normal Subgroup of General Linear Group

## Problem 332

Let $G=\GL(n, \R)$ be the **general linear group** of degree $n$, that is, the group of all $n\times n$ invertible matrices.

Consider the subset of $G$ defined by

\[\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.\]
Prove that $\SL(n, \R)$ is a subgroup of $G$. Furthermore, prove that $\SL(n,\R)$ is a normal subgroup of $G$.

The subgroup $\SL(n,\R)$ is called **special linear group**