{"id":"homomorphism-isomorphism","domain":"algebra","type":"Concept","title":"Homomorphism & Isomorphism","slug":"homomorphism-isomorphism","url":"/mathematics/algebra/homomorphism-isomorphism/","summary":"A homomorphism preserves algebraic structure between objects. An isomorphism is a perfectly reversible homomorphism, showing two structures are identical.","formal_definition":"phi(a*b)=phi(a)*phi(b) for homomorphism; bijective homomorphism = isomorphism","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Dummit, D.S. and Foote, R.M. (2004). Abstract Algebra. 3rd ed."},{"tier":1,"citation":"Lang, S. (2002). Algebra. Revised 3rd ed. Springer."},{"tier":2,"citation":"Mac Lane, S. (1998). Categories for the Working Mathematician. 2nd ed."},{"tier":3,"citation":"Pinter, C.C. (2010). A Book of Abstract Algebra. 2nd ed. Dover."}],"relationships":[{"type":"applies_to","target":"group","label":"Group"},{"type":"applies_to","target":"ring","label":"Ring"},{"type":"automorphisms_form","target":"galois-theory","label":"Galois Theory"},{"type":"used_to_classify","target":"symmetric-group","label":"Symmetric Group"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.9"}