Homomorphism between symmetric group and general linear group of order n.

Question

I am having trouble proving the following:

Show that $f: S_n \to GL_n(\mathbb{R}),\;\: f(x)=A_x$ is a homomorphism where $A_x$ is the permutation matrix associated with $x$.

$S_n$ is the symmetric group and $GL_n(\mathbb{R})$ is the general linear group of order $n$.

Answer 1

I can explicitly show you how to do this for $S_3$:

$e \to I$, the 3x3 identity matrix.

$(1\ 2) \to \begin{bmatrix}0&1&0\\1&0&0\\0&0&1 \end{bmatrix}$

$(1\ 3) \to \begin{bmatrix}0&0&1\\0&1&0\\1&0&0 \end{bmatrix}$

$(2\ 3) \to \begin{bmatrix}1&0&0\\0&0&1\\0&1&0 \end{bmatrix}$

$(1\ 2\ 3) \to \begin{bmatrix}0&0&1\\1&0&0\\0&1&0 \end{bmatrix}$

$(1\ 3\ 2) \to \begin{bmatrix}0&1&0\\0&0&1\\1&0&0 \end{bmatrix}$

This map assumes you multiply permutations "composition-wise" (right-to left). If you do it the other way, use the transposes.