Homomorphism Definition and Examples

Abstract Algebra · Axiom Academy

Understanding structure-preserving maps between groups through the fundamental property φ(ab) = φ(a)φ(b) This means we can either combine elements first then apply φ, or apply φ first then combine. The order doesn't matter—both paths give the same result. The sign homomorphism maps permutations to +1, −1 . Each permutation is classified as even (sign = +1) or odd (sign = −1) based on the number of transpositions needed to construct it. The determinant is a homomorphism from the group of invertible n×n matrices (under matrix multiplication) to the group of nonzero real numbers (under multiplication). 4. Example: Reduction Modulo n The reduction map takes integers to their remainders when divided by n. This is a homomorphism from (ℤ, +) to (ℤ n , +). 5. Operation Tables Under Homomorphisms When we have a homomorphism φ: G → H, the operation table of G must correspond to the operation table of H in a specific way. Let's visualize how the structure is preserved. This visualization shows how homomorphisms "compress" groups while maintaining their algebraic relationships. Every entry in the source group's table maps consistently to the target group's table.

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