Injectivity and Surjectivity

Abstract Algebra · Axiom Academy

LESSON Injective and Surjective Homomorphisms Understanding how the kernel determines injectivity and how the image relates to surjectivity in group homomorphisms. The kernel consists of all elements that map to the identity in H. For an injective homomorphism, only the identity element of G can map to the identity of H. 2. Visual Proof: φ Injective ⟺ Ker(φ) = e Theorem: A homomorphism φ is injective if and only if Ker(φ) = e_G . A homomorphism is surjective if and only if Im(φ) = H. That is, the image coincides with the entire codomain. 4. Classification of Homomorphisms Homomorphisms are classified by their injectivity and surjectivity properties:

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