Prove that $\varphi: S_n \to S_{n+1 }$, where for $\sigma\in S_{n}$: \begin{pmatrix} 1 & 2 & \dots & n \\ 1\sigma & 2\sigma & \dots & n\sigma\\ \end{pmatrix}
$\varphi(\sigma)\in S_{n+1}= \begin{pmatrix} 1 & 2 & \dots & n & n+1 \\ 1\sigma & 2\sigma & \dots & n\sigma & (n+1)\sigma \\ \end{pmatrix} $ is a homomorphism.
Wondering if anyone can explain how to prove this. Thanks.
I literally have no idea where to start, hence why I haven't given details of where ive got stuck as I have no idea what methods I should use to even start it. I have done a few questions similar using matrices, trigonometric functions and other functions and found them fine. I have been using the standard definition for homomorphism that is that φ(ab)=φ(a)φ(b). However, with this question this did not seem to get me very far at all and I was wondering if there was something more clever or a silly detail I was overlooking.