{"id":"symmetric-group","domain":"algebra","type":"Concept","title":"Symmetric Group","slug":"symmetric-group","url":"/mathematics/algebra/symmetric-group/","summary":"Sn = all n! rearrangements of n objects. Every finite group embeds in some Sn (Cayley's theorem). S5's non-solvability is why no quintic formula exists.","key_results":[{"result":"Cayley's theorem","statement":"Every group of order n embeds in Sn"},{"result":"Abel-Ruffini theorem","statement":"No radical formula for degree >=5 polynomials, since An is simple for n>=5"}],"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":"Fulton, W. and Harris, J. (1991). Representation Theory: A First Course. Springer."},{"tier":2,"citation":"Rotman, J.J. (1995). An Introduction to the Theory of Groups. 4th ed."},{"tier":3,"citation":"Stewart, I. (2015). Galois Theory. 4th ed."}],"relationships":[{"type":"is_a","target":"group","label":"Group"},{"type":"key_to","target":"galois-theory","label":"Galois Theory"},{"type":"classified_via","target":"homomorphism-isomorphism","label":"Cayley's theorem"},{"type":"related_to","target":"combinatorics","label":"Combinatorics — Young tableaux"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.9"}