How to disprove a group homomorphism?
$\text{For } n \in \mathbb N, \pi \in S_n \text{is } S_n \rightarrow S_n, \sigma \mapsto \pi \sigma \text{ a group homomorphism}$.
I would like to prove that this is wrong.
$\phi(xx´) = \pi \sigma \pi \sigma ´ \text{ and } \phi(x) \phi(x´)= \pi \sigma \pi \sigma´\text{ so: } \phi(xx´)= \phi(x) \phi(x´)$. Well, I see that I have proved the opposite but I have no idea how I could do it the right way.
A group homomorphism will always take the identity to the identity, but the given function takes the identity to $\pi$.