I have a question about the definition on this page.
I feel like conditions 1 and 2 imply 3, because a group homomorphism maps identity to identity.
Am I missing something? The proof of the lemma on that page uses condition 3 to assert that $1_F$ is not in the kernel of $\psi$, but I fail to see why this is not already implied by the first two conditions.
Finally, I would like to prove the lemma without the use of ideals. Could someone give me a hint as to how to show that the kernel of $\psi$ is trivial?