TITLE: Construction of an injective homomorphism from $S_n$ to $GL(n,\mathbb R)$
QUESTION [1 upvotes]: I am trying to define an injective homomorphism from $S_n$ to $GL(n,\mathbb R)$.

I simply don't have any idea how to start with. Any hint or suggestion will be appreciated.
If I define $f:S_n→GL(n,\mathbb R)$ by $f(σ)=A=(A_{ij})$, where $$A_{ij} = \left\{ \begin{array}{ll} 1  & \mbox{if } \sigma(j)=i \\ 0 & \mbox{if } \sigma(j)\ne i \end{array}\right.$$ will it work?
But the problem is how to show this is a homomorphism? Please help.

REPLY [0 votes]: Any action of a group  $G $ on a set $X $ gives a homomorphism from  $G $ to ${\rm Sym}(X) $.
So considering the action on the standard basis $X=\{e_1,e_2,\dots,e_n\} $ of $\mathbb R^n $,  it suffices to check that ${\rm Sym}(X)\le{\rm GL}(n,\mathbb R) $.
Indeed,  these correspond to  the so-called permutation matrices, with one $1$ in each row and column,  $0$'s everywhere else. Just take  $\sigma\in S_n $ to the transformation whose  matrix  has  $i $-th column equal to $e_{\sigma (i)} $.
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