Permutation matrix homomorphism

Question

Can someone please help me prove that permutation matrix is homomorphism? By that, I mean, let $f: S_n \to GL_n (\Bbb R), f(\sigma)=A_\sigma$ is homomorphism. The book tells me to prove it myself I have no idea how to. A small hint would do

Thanks in advance!

Answer 1

You need to show that, for all $\sigma_1, \sigma_2\in S_n$, $$ f(\sigma_1\sigma_2)=f(\sigma_1)f(\sigma_2). $$ In your case, this means

$$A_{\sigma_1\sigma_2}=A_{\sigma_1}A_{\sigma_2}.$$