Homomorphisms
Objective. Use kernels, images and isomorphisms.
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- 2. Worked example
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Learn: the key idea
Homomorphisms are maps φ: G → H between groups satisfying φ(ab) = φ(a)φ(b), preserving the group structure though not necessarily injectivity or surjectivity. The kernel ker(φ) = {g ∈ G : φ(g) = eₕ} is always a normal subgroup of G, and φ is injective exactly when ker(φ) is trivial. The image im(φ) is always a subgroup of H. For φ: ℤ → ℤ/5ℤ sending n to n mod 5, the kernel is 5ℤ, since those are exactly the integers mapping to the identity class 0 mod 5, and the image is all of ℤ/5ℤ since the map is surjective.
Worked example
Find the kernel of the homomorphism φ: (ℤ, +) → (ℤ/6ℤ, +) defined by φ(n) = n mod 6.
- 1The kernel consists of all integers n such that φ(n) equals the identity 0 mod 6.
- 2This means n mod 6 = 0, so n must be a multiple of 6.
- 3The set of such integers is exactly 6ℤ = {…, −6, 0, 6, 12, …}.
- 4Since this is non-trivial, φ is not injective, confirmed by the kernel size.
Answer: ker(φ) = 6ℤ
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Practice
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Start practisingCommon mistake. Students assume a homomorphism's kernel can be any subset; the kernel must always contain the identity and be closed under the group operation and inverses, so sets like {1, 2} are never valid kernels in multiplicative groups unless they form a subgroup.
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