Verifying φ

Abstract Algebra · Axiom Academy

EXAMPLE Verifying φ: Z → Z₅ is a Homomorphism Step-by-step verification that reduction modulo n preserves the group operation The map φ groups infinitely many integers into 5 equivalence classes: Notice: All our highlighted values (7, 13, 20) from Z map to elements (2, 3, 0) in Z₅, and the operation is preserved! Excellent work! You've verified that φ: Z → Zₙ is indeed a homomorphism. Here's what we learned: Homomorphism Property: φ(a + b) = φ(a) + φ(b) means the function preserves the group operation - we can add before or after applying φ Two Paths, Same Result: Whether we add in Z then reduce mod n, or reduce mod n then add in Zₙ, we get the same answer Why It Works: Modular arithmetic is compatible with addition - the remainder of a sum equals the sum of remainders (mod n) Visual Insight: φ maps the infinite group Z onto the finite group Zₙ, collapsing integers into n equivalence classes while preserving structure General Principle: This works for ANY n and ANY pair of integers a, b - not just our specific example! This reduction map is fundamental in abstract algebra. It shows how we can study infinite groups by examining their finite quotients, and it's the foundation for understanding quotient groups and the First Isomorphism Theorem!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.