Math 2.0

Group theory and generalizations Definition

Torsion subgroup

Builds on Abelian groupSubgroupOrder of an elementQuotient group

Definition

Let AA be an abelian group. The torsion subgroup of AA is AT={a∈A:a has finite order}A_T = \{a \in A : a \text{ has finite order}\}. The group AA is a torsion group when AT=AA_T = A, and it is torsion-free when AT={e}A_T = \{e\}.

Remarks

The set ATA_T is a subgroup because AA is commutative: if am=ea^m = e and bn=eb^n = e, then (ab)mn=e(a b)^{m n} = e. In a nonabelian group, the elements of finite order need not form a subgroup. In ⟨x,y∣x2,y2⟩\langle x, y \mid x^2, y^2 \rangle, the product xyx y has infinite order. The quotient group A/ATA / A_T is torsion-free, and ATA_T is a characteristic subgroup of AA. For A=ℚ/ℤA = \mathbb{Q}/\mathbb{Z}, each element has finite order, so AT=AA_T = A, but the orders have no upper bound.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionTorsion subgroup, definition, no evidence levelTorsion subgroupdefinition
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  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Conjugate elements definition
  5. Laws of exponents in a group statement
  6. Subgroup definition
  7. Abelian group definition
  8. Order of an element definition
  9. Conjugate subgroup definition
  10. Left coset definition
  11. Right coset definition
  12. Normal subgroup definition
  13. Normality via left and right cosets statement
  14. Quotient group definition
  15. Torsion subgroup definition

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Torsion_subgroup, lead section
  2. Web https://en.wikipedia.org/wiki/Torsion_subgroup, Examples and further results
About this page
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Kinddefinition
WritingW1 stated
MSC20K10
DefinesTorsion subgroup
UsesAbelian group, Subgroup, Order of an element, Quotient group

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