Math 2.0

Group theory and generalizations Definition

Orderable group

Builds on GroupTotal orderLinearly ordered groupTorsion-free groupAbelian groupFree group

Definition

Let GG be a group. The group GG is orderable, or bi-orderable, if there is a total order ≤\le on GG such that for all a,b,c∈Ga, b, c \in G, a≤b  ⟹  ca≤cb and ac≤bc.a \le b \implies ca \le cb \text{ and } ac \le bc. That is, (G,≤)(G, \le) is a linearly ordered group for some order ≤\le.

Remarks

Example: ℤ2\Z^2 under addition is orderable, since the lexicographic order is total and translation invariant. Non-example: the cyclic group ℤ/2ℤ\Z/2\Z is not orderable. If gg is its nonzero element and 0<g0 < g, then adding gg gives g<g+g=0g < g + g = 0, a contradiction; the case g<0g < 0 is the same. More generally, an orderable group is a torsion-free group. Every torsion-free abelian group is orderable. Every free group is orderable. A group is orderable if and only if it has a subset PP with PP⊆PPP \subseteq P, gPg−1=PgPg^{-1} = P for each g∈Gg \in G, and G=P⊔{e}⊔P−1G = P \sqcup \set{e} \sqcup P^{-1}.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first.

  1. Binary operation definition
  2. Element definition
  3. Free group concept
  4. Group concept
  5. Integers object
  6. Order of an element concept
  7. Proposition concept
  8. Rational numbers object
  9. Reduced word definition
  10. Torsion subgroup concept
  11. Commutativity definition
  12. Set definition
  13. Commutator of two elements definition
  14. Conjugate elements definition
  15. Direct product of groups definition
  16. Group homomorphism definition
  17. Laws of exponents in a group statement
  18. Order of a group definition
  19. Subgroup definition
  20. Implication definition
  21. Predicate definition
  22. Ordered pair definition
  23. Abelian group definition
  24. Order of an element definition
  25. Group isomorphism definition
  26. Conjugate subgroup definition
  27. Cyclic subgroup generated by an element definition
  28. Left coset definition
  29. Right coset definition
  30. Subgroup criterion statement
  31. Universal quantifier definition
  32. Cartesian product definition
  33. g to the n equals 1 iff the order divides n statement
  34. Cyclic group definition
  35. Subset definition
  36. Free group definition
  37. Normal subgroup definition
  38. Intersection of subgroups is a subgroup statement
  39. Binary relation definition
  40. Order of a cyclic subgroup equals the order of its generator statement
  41. Normality via left and right cosets statement
  42. Subgroup generated by a subset definition
  43. Antisymmetric relation definition
  44. Reflexive relation definition
  45. Transitive relation definition
  46. Commutator subgroup definition
  47. Finitely generated group definition
  48. Cyclic groups of the same order are isomorphic statement
  49. Quotient group definition
  50. Preorder definition
  51. Product of cyclic groups is cyclic iff the orders are coprime statement
  52. Torsion subgroup definition
  53. Abelianisation definition
  54. Partial order definition
  55. Classification of finite abelian groups statement
  56. Free abelian group definition
  57. Lexicographic order definition
  58. Total order definition
  59. Ordered group definition
  60. Structure theorem for finitely generated abelian groups statement
  61. Linearly ordered group definition
  62. Torsion-free group definition
  63. Orderable group definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Partially_ordered_group
About this page
TagZZFFBXDTWN
URIm2://ZZFFBXDTWN
Kinddefinition
WritingW0 stub
MSC20F60, 06F15
DefinesOrderable group
UsesGroup, Total order, Linearly ordered group, Torsion-free group, Abelian group, Free group
Changed2 November 2026

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