Math 2.0

Group theory and generalizations Definition

Torsion-free group

Builds on GroupOrder of an elementIntegersRational numbersFree groupTorsion subgroupStructure theorem for finitely generated abelian groups

Definition

Let GG be a group with identity ee. GG is torsion-free if every element g≠eg \ne e of GG has infinite order, that is, gn≠eg^n \ne e for all integers n≥1n \ge 1.

Remarks

Example: the integers ℤ\Z under addition. If m≠0m \ne 0 and n≥1n \ge 1, then nm≠0n m \ne 0. The groups ℤk\Z^k, the rational numbers under addition, and free groups are also torsion-free.

Non-example: ℤ×ℤ/2ℤ\Z \times \Z/2\Z. The element (0,1)(0, 1) is not the identity, but 2⋅(0,1)=(0,0)2 \cdot (0, 1) = (0, 0).

For an abelian group AA with torsion subgroup TT, the quotient A/TA/T is torsion-free. By the structure theorem for finitely generated abelian groups, a finitely generated torsion-free abelian group is isomorphic to ℤn\Z^n for some n≥0n \ge 0. In a nonabelian group the elements of finite order need not form a subgroup. In the infinite dihedral group, two reflections have order 22, but their product is a translation of infinite order.

Reading path

To understand this page from the ground up, read these 43 pages in order. The foundations come first.

  1. Binary operation definition
  2. Free group concept
  3. Group concept
  4. Integers object
  5. Order of an element concept
  6. Rational numbers object
  7. Reduced word definition
  8. Torsion subgroup concept
  9. Commutativity definition
  10. Commutator of two elements definition
  11. Conjugate elements definition
  12. Direct product of groups definition
  13. Group homomorphism definition
  14. Laws of exponents in a group statement
  15. Order of a group definition
  16. Subgroup definition
  17. Abelian group definition
  18. Order of an element definition
  19. Group isomorphism definition
  20. Conjugate subgroup definition
  21. Cyclic subgroup generated by an element definition
  22. Left coset definition
  23. Right coset definition
  24. Subgroup criterion statement
  25. g to the n equals 1 iff the order divides n statement
  26. Cyclic group definition
  27. Free group definition
  28. Normal subgroup definition
  29. Intersection of subgroups is a subgroup statement
  30. Order of a cyclic subgroup equals the order of its generator statement
  31. Normality via left and right cosets statement
  32. Subgroup generated by a subset definition
  33. Commutator subgroup definition
  34. Finitely generated group definition
  35. Cyclic groups of the same order are isomorphic statement
  36. Quotient group definition
  37. Product of cyclic groups is cyclic iff the orders are coprime statement
  38. Abelianisation definition
  39. Torsion subgroup definition
  40. Classification of finite abelian groups statement
  41. Free abelian group definition
  42. Structure theorem for finitely generated abelian groups statement
  43. Torsion-free group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Torsion_(algebra)
About this page
TagVQ4SSKS53D
URIm2://VQ4SSKS53D
Kinddefinition
WritingW0 stub
MSC20F99, 20K15
DefinesTorsion-free group
UsesGroup, Order of an element, Integers, Rational numbers, Free group, Torsion subgroup, Structure theorem for finitely generated abelian groups
Changed2 November 2026

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