Homomorphism
★★★★☆Undergraduate
📖Definition
A homomorphism is a function between algebraic structures that preserves operations. It's the key tool for comparing and classifying groups, rings, fields, etc.
📐Formulas
φ(a · b) = φ(a) · φ(b)
Group homomorphism definition
φ(a + b) = φ(a) + φ(b), φ(ab) = φ(a)φ(b)
Ring homomorphism definition
ker(φ) = a : φ(a) = e
Kernel
im(φ) = φ(a) : a ∈ G
Image
✏️Examples
예제 1
Show exp: (ℝ, +) → (ℝ⁺, ×) is a homomorphism.
예제 2
What is the kernel of φ: ℤ → ℤₙ, φ(a) = a mod n?
⚡Applications
Algebra
Structure classification, isomorphism theorems
Cryptography
Group-based protocols
Physics
Symmetry group representations
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