← MATH 01340 · Modern Algebra I
MATH 01340 · Groups
Homomorphisms / isomorphisms
Definition
Homomorphism φ:G→H preserves the operation: φ(ab)=φ(a)φ(b). Iso = bijective homo (same structure). Kernel and image encode structure (1st iso theorem).
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Kernel is always a normal subgroup; image is a subgroup of the codomain.
Examples: det: GLₙ→ℝˣ; mod n map ℤ→ℤₙ.
Isomorphic groups are “the same” abstractly — classify by finding invariants.
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1.Show φ:ℤ→ℤₙ, φ(x)=x mod n is a homomorphism and find ker φ.