Group homomorphisms preserve identity elements.

Group homomorphisms preserve inverses.

Let \(f: G \longrightarrow H\) be a group homomorphism. The \(f\) is injective if and only if \(\ker f = e_G.\)

The fact that group homomorphisms preserve identity elements is also proven in:
I don't think that video proves inverses are preserved, because he only proves the theorem for right inverses. Either way, his proofs are essentially ProofWiki's but more terse. I prefer more verbose proofs when dealing with objects just introduced, so I won't list the video above on this page.