{
  "id": "group",
  "domain": "algebra",
  "type": "Concept",
  "title": "Group",
  "slug": "group",
  "url": "/mathematics/algebra/group/",
  "summary": "A set with a binary operation satisfying closure, associativity, identity, and invertibility. The fundamental structure of abstract algebra; the mathematical language of symmetry.",
  "formal_definition": "(G, ★) is a group if ★ is associative, there exists an identity e ∈ G, and every element has an inverse. Closure is implicit in ★ being a binary operation on G.",
  "key_results": [
    { "result": "Lagrange's Theorem", "statement": "|H| divides |G| for every subgroup H of a finite group G" },
    { "result": "First Isomorphism Theorem", "statement": "G / ker(φ) ≅ im(φ) for any homomorphism φ : G → H" },
    { "result": "Classification of Finite Simple Groups", "statement": "Every finite simple group is cyclic, alternating, of Lie type, or one of 26 sporadic groups. Proof completed 2004." }
  ],
  "introduced_by": "Évariste Galois, c. 1830 (term 'group' / 'groupe')",
  "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. John Wiley & Sons.", "chapters": "1–6" },
    { "tier": 1, "citation": "Lang, S. (2002). Algebra. Revised 3rd ed. Springer (Graduate Texts in Mathematics, 211)." },
    { "tier": 2, "citation": "Rotman, J.J. (1995). An Introduction to the Theory of Groups. 4th ed. Springer." },
    { "tier": 3, "citation": "Stewart, I. (2015). Galois Theory. 4th ed. CRC Press." }
  ],
  "relationships": [
    { "type": "generalised_by", "target": "ring", "label": "Ring" },
    { "type": "generalised_by", "target": "field", "label": "Field" },
    { "type": "key_person", "target": "galois-evariste", "label": "Évariste Galois" },
    { "type": "key_person", "target": "noether-emmy", "label": "Emmy Noether" }
  ],
  "entry_status": "live",
  "reading_levels_available": ["curious", "exploring", "deep_dive"],
  "completeness": 1.0,
  "codex_version": "1.1"
}