Math 2.0

Group theory and generalizations Definition

Free abelian group

Builds on Abelian groupGroup isomorphismAbelianisationFree group

Definition

Let AA be an abelian group, written additively. A basis of AA is a subset B⊆AB \subseteq A such that each a∈Aa \in A has exactly one expression a=k1b1+⋯+knbna = k_1 b_1 + \cdots + k_n b_n with distinct bi∈Bb_i \in B and nonzero integers kik_i. The group AA is a free abelian group when it has a basis.

Remarks

The additive group ℤ\mathbb{Z} is free abelian with basis {1}\{1\}, and ℤ2\mathbb{Z}^2 is free abelian with basis {(1,0),(0,1)}\{(1, 0), (0, 1)\}. All bases of AA have the same cardinality, the rank, and a free abelian group of finite rank nn is isomorphic to ℤn\mathbb{Z}^n. The free abelian group on a set SS is the abelianisation of the free group FSF_S. Each subgroup of a free abelian group is free abelian.

Reading path

To understand this page from the ground up, read these 23 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 elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementFree group, definition, no evidence levelFree groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…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 subgroupdefinitionQuotient group, definition, no evidence levelQuotient groupdefinitionAbelianisation, definition, no evidence levelAbelianisationdefinitionFree abelian group, definition, no evidence levelFree abelian groupdefinition
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. Group homomorphism definition
  8. Subgroup definition
  9. Abelian group definition
  10. Conjugate subgroup definition
  11. Group isomorphism definition
  12. Left coset definition
  13. Right coset definition
  14. Subgroup criterion statement
  15. Free group definition
  16. Normal subgroup definition
  17. Intersection of subgroups is a subgroup statement
  18. Normality via left and right cosets statement
  19. Subgroup generated by a subset definition
  20. Commutator subgroup definition
  21. Quotient group definition
  22. Abelianisation definition
  23. Free abelian group definition

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Free_abelian_group, Definition and examples
  2. Web https://en.wikipedia.org/wiki/Free_abelian_group, Rank
  3. Web https://en.wikipedia.org/wiki/Free_abelian_group, Subgroups
  4. Web https://en.wikipedia.org/wiki/Free_group, Free abelian group
About this page
Tag0YMSEZE1VH
URIm2://0YMSEZE1VH
Kinddefinition
WritingW1 stated
MSC20K15, 20K20
DefinesFree abelian group
UsesAbelian group, Group isomorphism, Abelianisation, Free group

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