Question: Let $\sigma\in S_n$ be a permutation. We define the permutation matrix $P(\sigma)$ as the $n\times n$ matrix:
$$
P(\sigma)_{ij}=
\begin{cases}
1&i=\sigma(j)\\
0&i\neq\sigma(j)
\end{cases}
$$
We see that $P(\sigma)$ permutes the columns of the identify matrix according to permutation $\sigma$. The mapping $P\colon S_n\to GL_n(N,\mathbb R)\colon\sigma\mapsto P(\sigma)$ is an injective groupsmorphism.

@Astyx I don't see where I make a mistake though;
$$
\begin{pmatrix}
0&0&1\\
0&1&0\\
1&0&0
\end{pmatrix}
\cdot
\begin{pmatrix}
1&0&0\\
0&0&1\\
0&1&0
\end{pmatrix}=
\begin{pmatrix}
0&1&0\\
0&0&1\\
1&0&0
\end{pmatrix}.
$$
This matrix corresponds with $\sigma=(123)$, while we need $\sigma=(132)$.

@Astyx Ah right, I had no idea I could look at it that way. However, my book states that the columns of the identity matrix are permuted according to the permutation. Why can't I just look at how those columns are permuted? It seems to me that $(321)$ corresponds with
$$
\begin{pmatrix}
0&0&1\\
1&0&0\\
0&1&0
\end{pmatrix}
$$