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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.

Now I define $f$ as commented bellow. But the problem is how to show this is a homomorphism? Please help.

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1 Answer 1

Hint: Let $S_n$ act on the $n$ element set given by the standard basis of $\mathbb R^n$.

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If I define $f:S_n\to GL(n,\mathbb R)$ by $f(\sigma)=A_{ij}$, $where, A_{ij} = \{ \begin{array}{ll} 1 & \mbox{if } \sigma(j)=i \\ 0 & \mbox{if } \sigma(j)\ne i \end{array}$ Will it work? –  Anupam Oct 18 '13 at 16:23
    
Sorry, $f(\sigma)=A=(A_{ij})$ –  Anupam Oct 18 '13 at 16:35

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