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Notes for page 26

A kernel of f is the set \[\{ x ∈ G | f(x) = e’ \}\] An injective homomorphism \(f:G → G’\) is called an embedding.

A homomorphism whose kernel is trivial is injective.

Proof: Given \(f(x) = f(y)\), then \(f(xy^-1) = f(x)f(y^-1) = e’\) Therefore \(xy^-1 = e\) and \(x = y\)