Abelian Group Ontology

FLOW ALGEBRA
TITLE: Abelian Group Ontology
VERSION: 4.0
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/algebraic-entities/abelian-group-ontology-3.html
SOURCE_ALIGNMENT: Independent mathematical construction; source page not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
DATE: 2026-10-13

01. ENTITY DEFINITION
ENTITY AbelianGroup := (G, operation, identity, inverse)
TYPE: AlgebraicStructure > Group > AbelianGroup
DOMAIN: AbstractAlgebra
DEFINITION: A group (G, *) is Abelian precisely when a*b = b*a for every a,b in G.
ALIAS: commutative group

02. SIGNATURE AND TYPES
G: nonempty Set
operation: G x G -> G
identity: G
inverse: G -> G
member(a,G): Boolean
subgroup(H,G): Boolean
normalSubgroup(H,G): Boolean
homomorphism(f,G,K): Boolean
isomorphism(f,G,K): Boolean
order(G): Cardinal
orderOf(a): PositiveInteger or Infinity

03. AXIOMS
A01 CLOSURE: forall a,b in G, a*b in G.
A02 ASSOCIATIVITY: forall a,b,c in G, (a*b)*c = a*(b*c).
A03 IDENTITY: exists e in G, forall a in G, e*a = a*e = a.
A04 INVERSES: forall a in G, exists b in G, a*b = b*a = e.
A05 COMMUTATIVITY: forall a,b in G, a*b = b*a.
A06 NONEMPTY: G != empty set.
A07 TOTAL_OPERATION: operation defined for all ordered pairs in G x G.

04. SEMANTIC RELATIONSHIPS
AbelianGroup IS_A Group.
AbelianGroup IS_A AlgebraicStructure.
AbelianGroup HAS_PROPERTY Commutativity.
AbelianGroup HAS_OPERATION BinaryOperation.
AbelianGroup HAS_ELEMENT IdentityElement.
AbelianGroup HAS_MAP InverseMap.
Subgroup OF AbelianGroup IS_A AbelianGroup.
AbelianGroup IS_RELATED_TO Module.
AbelianGroup IS_RELATED_TO IntegerModule.
AbelianGroup SUPPORTS QuotientGroup.
AbelianGroup SUPPORTS DirectProduct.
AbelianGroup SUPPORTS GroupHomomorphism.
AbelianGroup CONTRASTS_WITH NonAbelianGroup.

05. CANONICAL EXAMPLES
E01 (Z,+) is Abelian; identity 0, inverse of n is -n.
E02 (Q,+), (R,+), (C,+) are Abelian.
E03 (Z/nZ,+) is finite cyclic Abelian for integer n >= 1.
E04 (R without 0, multiplication) is Abelian.
E05 (C without 0, multiplication) is Abelian.
E06 ({1,-1}, multiplication) is Abelian.
E07 (Z/2Z x Z/2Z, componentwise addition) is Abelian but not cyclic.
E08 Any vector space under vector addition is an Abelian group.
E09 Every cyclic group is Abelian.
E10 A direct product of Abelian groups is Abelian.
E11 The trivial one-element group is Abelian.
E12 The group of invertible diagonal n x n real matrices under multiplication is Abelian.

06. NON-EXAMPLES AND BOUNDARIES
N01 Symmetric group S3 is not Abelian.
N02 General linear group GL(2,R) is not Abelian.
N03 Matrix multiplication on all square matrices is not generally commutative and not a group on all matrices.
N04 (N,+) is not a group: additive inverses are missing.
N05 (Z,multiplication) is not a group: most elements have no multiplicative inverse.
N06 Commutativity alone does not imply group axioms.

07. DERIVED IDENTITIES
D01 Identity element is unique.
D02 Every element has a unique inverse.
D03 (a*b)^-1 = a^-1*b^-1 (also holds in reversed order by commutativity).
D04 For integers m,n, a^m*a^n = a^(m+n).
D05 For integers n, (a*b)^n = a^n*b^n.
D06 Every subgroup of an Abelian group is normal.
D07 Every quotient of an Abelian group is Abelian.
D08 The centre Z(G) of an Abelian group equals G.
D09 Every commutator a*b*a^-1*b^-1 equals e.
D10 The commutator subgroup [G,G] is trivial.
D11 The direct product of any family of Abelian groups is Abelian.
D12 The image of an Abelian group under a homomorphism is Abelian.
D13 Every subgroup of an Abelian group is Abelian.
D14 The kernel of a homomorphism is a normal subgroup.
D15 An Abelian group can be viewed as a Z-module.

08. FORMAL INFERENCE RULES
R01 IF group(G) AND forall a,b in G, a*b=b*a THEN abelian(G).
R02 IF abelian(G) THEN group(G).
R03 IF abelian(G) AND subgroup(H,G) THEN abelian(H).
R04 IF abelian(G) AND subgroup(H,G) THEN normalSubgroup(H,G).
R05 IF abelian(G) AND normalSubgroup(H,G) THEN abelian(G/H).
R06 IF cyclic(G) THEN abelian(G).
R07 IF abelian(G) AND homomorphism(f,G,K) THEN abelian(image(f)).
R08 IF abelian(G) AND isomorphism(f,G,K) THEN abelian(K).
R09 IF abelian(G) AND abelian(H) THEN abelian(G x H).
R10 IF abelian(G) THEN centre(G)=G.
R11 IF abelian(G) THEN commutatorSubgroup(G)={e}.
R12 IF group(G) AND exists a,b in G with a*b != b*a THEN NOT abelian(G).
R13 IF finite(G) AND order(G) is prime THEN cyclic(G) AND abelian(G).
R14 IF group(G) AND order(G)=1 THEN abelian(G).
R15 IF abelian(G) THEN every inner automorphism of G is identity.
R16 IF abelian(G) AND finite(G) THEN G is isomorphic to a finite direct product of cyclic groups of prime-power order.
R17 IF finitelyGenerated(G) AND abelian(G) THEN G is isomorphic to Z^r x finiteAbelianTorsionGroup for some r>=0.
R18 IF abelian(G) AND a,b in G THEN (a*b)^n=a^n*b^n for integer n.
R19 IF abelian(G) AND a,b in G THEN a*b*a^-1*b^-1=e.
R20 IF group(G) AND every element commutes with every element THEN abelian(G).
R21 IF abelian(G) AND subgroup(H,G) THEN quotientGroup(G,H) is defined.
R22 IF abelian(G) AND homomorphism(f,G,K) THEN kernel(f) is normal in G.
R23 IF abelian(G) THEN additiveNotationPreferred(G) is conventional, not mandatory.
R24 IF abelian(G) AND subgroup(H,G) AND subgroup(K,G) THEN H+K is a subgroup of G (additive notation).
R25 IF abelian(G) AND subgroup(H,G) AND subgroup(K,G) THEN H intersect K is a subgroup of G.
R26 IF abelian(G) AND finite(G) THEN every subgroup order divides |G|.
R27 IF abelian(G) AND a in G THEN generatedSubgroup(a) is cyclic and Abelian.
R28 IF abelian(G) AND homomorphism(f,G,K) THEN G/kernel(f) is isomorphic to image(f).
R29 IF abelian(G) AND n is integer THEN map a -> a^n is a group endomorphism.
R30 IF abelian(G) THEN G has a canonical Z-module structure via integer scalar multiplication.

09. FINITE ABELIAN GROUP CLASSIFICATION
THEOREM: Every finite Abelian group is isomorphic to a direct product of cyclic groups of prime-power order.
PRIMARY_FORM: G ~= product over primes p of product_i Z/(p^k_i)Z.
INVARIANT_FACTOR_FORM: G ~= Z/n1Z x ... x Z/nrZ where n1 divides n2 divides ... divides nr.
EXAMPLE: Every Abelian group of order 8 is isomorphic to exactly one of Z/8Z, Z/4Z x Z/2Z, or (Z/2Z)^3.
LIMITATION: The classification applies to finite Abelian groups; it is not a classification of all infinite Abelian groups.

10. FINITELY GENERATED ABELIAN GROUPS
THEOREM: Every finitely generated Abelian group G ~= Z^r x Z/n1Z x ... x Z/nkZ, with n_i>=2 and n_i dividing n_(i+1).
RANK: r is uniquely determined.
TORSION: Finite-order elements form a subgroup Tor(G).
EXAMPLE: Z^2 x Z/6Z has free rank 2 and torsion subgroup Z/6Z.

11. OPERATIONS AND TRANSFORMATIONS
OP01 direct_product(G,H) -> AbelianGroup when both inputs Abelian.
OP02 quotient(G,H) -> AbelianGroup when H subgroup of Abelian G.
OP03 subgroup_generated_by(G,S) -> AbelianGroup for S subset G.
OP04 homomorphic_image(f,G) -> AbelianGroup if G Abelian.
OP05 integer_scalar_action(n,a) -> a^n (multiplicative) or n*a (additive).
OP06 compute_centre(G) -> G if G Abelian.
OP07 compute_commutator(a,b) -> e if G Abelian.

12. VALIDATION CONSTRAINTS
V01 Check closure for operation on carrier set.
V02 Check associativity for every triple or by a valid proof.
V03 Check existence of a two-sided identity.
V04 Check existence of a two-sided inverse for each element.
V05 Check commutativity for every pair or by a valid proof.
V06 Reject structures failing any group axiom even if commutative.
V07 Distinguish Abelian group from Abelian semigroup or commutative monoid.
V08 Require explicit operation: a set alone does not determine a group.
V09 Treat a finite Cayley table as evidence only after checking completeness and all axioms.
V10 Do not infer cyclicity from Abelian property.
V11 Do not infer finiteness from Abelian property.
V12 Do not infer that all groups are Abelian.
V13 Preserve distinction between abstract isomorphism and literal equality.
V14 For infinite groups, do not apply finite classification without hypotheses.
V15 For quotient groups, verify normality (automatic for subgroups of Abelian groups).

13. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE("Abelian group", "commutative group") = exact.
SEMANTIC_EQUIVALENCE("additive Abelian group", "Abelian group written additively") = contextual.
SEMANTIC_EQUIVALENCE("Z-module", "Abelian group") = equivalence of categories with canonical correspondence.
SEMANTIC_EQUIVALENCE("cyclic group", "Abelian group") = false (cyclic implies Abelian, not converse).
SEMANTIC_EQUIVALENCE("commutative monoid", "Abelian group") = false (inverses may be absent).

14. KNOWLEDGE GRAPH TRIPLES
(AbelianGroup, subclassOf, Group)
(AbelianGroup, hasProperty, Commutativity)
(AbelianGroup, hasAxiom, Associativity)
(AbelianGroup, hasAxiom, Identity)
(AbelianGroup, hasAxiom, Inverse)
(CyclicGroup, subclassOf, AbelianGroup)
(AbelianGroup, equivalentCategoryTo, ZModule)
(SubgroupOfAbelianGroup, isNormalIn, ParentAbelianGroup)
(QuotientOfAbelianGroup, instanceOf, AbelianGroup)
(FiniteAbelianGroup, classifiedBy, FundamentalTheoremOfFiniteAbelianGroups)
(FinitelyGeneratedAbelianGroup, classifiedBy, StructureTheoremForFinitelyGeneratedAbelianGroups)

15. CONFIDENCE
CONFIDENCE mathematical_axioms: 1.00
CONFIDENCE standard_theorems: 1.00
CONFIDENCE worked_examples: 0.99
CONFIDENCE source_page_alignment: unverified
CONFIDENCE notes: Confidence values express formal mathematical certainty or editorial assessment, not measured probabilistic calibration.

16. UPDATE_MODEL
UPDATE_MODEL version: 4.0
UPDATE_MODEL revision_policy: preserve axioms; extend examples, relations and verified source mappings.
UPDATE_MODEL validation: require mathematical proof or authoritative reference for substantive theorem changes.
UPDATE_MODEL provenance: retain source URL and distinguish independently generated content from verified page content.
UPDATE_MODEL compatibility: preserve stable entity IDs and typed relations across releases.

17. FLOW_HASH
FLOW_HASH algorithm: SHA-256
FLOW_HASH scope: canonical UTF-8 bytes of this file with the FLOW_HASH_VALUE line excluded.
FLOW_HASH_VALUE: COMPUTE_ON_EXPORT

18. AI_CONSUMPTION
AI_CONSUMPTION parse_entity: AbelianGroup
AI_CONSUMPTION require_axioms: [closure, associativity, identity, inverse, commutativity]
AI_CONSUMPTION inference_mode: typed, theorem-constrained
AI_CONSUMPTION notation: additive or multiplicative, consistently declared
AI_CONSUMPTION distinguish: [Group, AbelianGroup, CyclicGroup, CommutativeMonoid]
AI_CONSUMPTION reject_unsupported_inferences: true
AI_CONSUMPTION provenance_required: true
AI_CONSUMPTION classification_guard: apply finite or finitely generated theorems only under their hypotheses
AI_CONSUMPTION output: validated entities, relations, theorems, examples, confidence and provenance

END FLOW ALGEBRA