Math 2.0

Group theory and generalizations Abelian groups Theorem

Structure theorem for finitely generated abelian groups

Builds on Finitely generated groupAbelian groupFree abelian groupTorsion subgroupClassification of finite abelian groups

Statement

Let GG be a finitely generated abelian group. Then G≅ℤn⊕ℤ/q1ℤ⊕⋯⊕ℤ/qtℤG \cong \mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z} for an integer n≥0n \geq 0 and prime powers q1,…,qtq_1, \ldots, q_t. Also G≅ℤn⊕ℤ/k1ℤ⊕⋯⊕ℤ/kuℤG \cong \mathbb{Z}^n \oplus \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z} with 1<k1∣k2∣⋯∣ku1 < k_1 \mid k_2 \mid \cdots \mid k_u. The rank nn, the multiset of the qiq_i and the sequence of the kik_i are unique.

Remarks

The finite part is the torsion subgroup of GG, and ℤn\mathbb{Z}^n is a free abelian group of rank nn. So GG is finite if and only if n=0n = 0, and then the theorem is the classification of finite abelian groups. Each finitely generated torsion-free abelian group is free abelian. The group (ℚ,+)(\mathbb{Q}, +) shows that the condition "finitely generated" is necessary. The theorem is a special case of the structure theorem for finitely generated modules over a principal ideal domain.

History

Henry John Stephen Smith solved the case of finitely presented abelian groups in 1861, with the Smith normal form of integer matrices. Henri Poincaré proved the theorem for finitely generated abelian groups in 1900, with a matrix proof. Emmy Noether extended Kronecker's proof to finitely generated abelian groups in 1926.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptReduced word, definition, no evidence levelReduced worddefinitionCommutativity, definition, no evidence levelCommutativitydefinitionCommutator of two elements, definition, no evidence levelCommutator of two ele…definitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionDirect product of groups, definition, no evidence levelDirect product of gro…definitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementOrder of a group, definition, no evidence levelOrder of a groupdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementFree group, definition, no evidence levelFree groupdefinitiong to the n equals 1 iff the order divides n, statement, no evidence levelg to the n equals 1 i…statementCyclic group, definition, no evidence levelCyclic groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementOrder of a cyclic subgroup equals the order of its generator, statement, no evidence levelOrder of a cyclic sub…statementNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionCommutator subgroup, definition, no evidence levelCommutator subgroupdefinitionFinitely generated group, definition, no evidence levelFinitely generated gr…definitionCyclic groups of the same order are isomorphic, statement, no evidence levelCyclic groups of the …statementQuotient group, definition, no evidence levelQuotient groupdefinitionAbelianisation, definition, no evidence levelAbelianisationdefinitionProduct of cyclic groups is cyclic iff the orders are coprime, statement, no evidence levelProduct of cyclic gro…statementTorsion subgroup, definition, no evidence levelTorsion subgroupdefinitionFree abelian group, definition, no evidence levelFree abelian groupdefinitionClassification of finite abelian groups, statement, no evidence levelClassification of fin…statementStructure theorem for finitely generated abelian groups, statement, no evidence levelStructure theorem for…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Reduced word definition
  4. Commutativity definition
  5. Commutator of two elements definition
  6. Conjugate elements definition
  7. Direct product of groups definition
  8. Group homomorphism definition
  9. Laws of exponents in a group statement
  10. Order of a group definition
  11. Subgroup definition
  12. Abelian group definition
  13. Order of an element definition
  14. Group isomorphism definition
  15. Conjugate subgroup definition
  16. Cyclic subgroup generated by an element definition
  17. Left coset definition
  18. Right coset definition
  19. Subgroup criterion statement
  20. Free group definition
  21. g to the n equals 1 iff the order divides n statement
  22. Cyclic group definition
  23. Normal subgroup definition
  24. Intersection of subgroups is a subgroup statement
  25. Order of a cyclic subgroup equals the order of its generator statement
  26. Normality via left and right cosets statement
  27. Subgroup generated by a subset definition
  28. Commutator subgroup definition
  29. Finitely generated group definition
  30. Cyclic groups of the same order are isomorphic statement
  31. Quotient group definition
  32. Abelianisation definition
  33. Product of cyclic groups is cyclic iff the orders are coprime statement
  34. Torsion subgroup definition
  35. Free abelian group definition
  36. Classification of finite abelian groups statement
  37. Structure theorem for finitely generated abelian groups statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Finitely_generated_abelian_group, Primary decomposition
  2. Web https://en.wikipedia.org/wiki/Finitely_generated_abelian_group, Invariant factor decomposition
  3. Web https://en.wikipedia.org/wiki/Finitely_generated_abelian_group, Corollaries
  4. Web https://en.wikipedia.org/wiki/Finitely_generated_abelian_group, History
About this page
Tag0RY0B72XAS
URIm2://0RY0B72XAS
Aliasesfundamental-theorem-of-finitely-generated-abelian-groups
Kindstatement, theorem
WritingW0 stub
MSC20K25
StatusThis copy of the site has no build data. Run m2 status 0RY0B72XAS.
UsesFinitely generated group, Abelian group, Free abelian group, Torsion subgroup, Classification of finite abelian groups

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