Math 2.0

Group theory and generalizations Proof

Proof that groups of prime order are cyclic

Builds on Order of an element divides the order of the groupPrime numberCyclic subgroup generated by an elementCyclic group

Proof

  1. Let GG be a group of order pp, where pp is a prime number.
  2. Since p≥2p \ge 2, GG contains an element gg that is not the identity ee.
  3. The order of gg divides pp. Thus the order of gg is 11 or pp.
  4. Only the identity has order 11, and g≠eg \ne e. Thus gg has order pp.
  5. Thus the cyclic subgroup ⟨g⟩\langle g \rangle has pp elements, so ⟨g⟩=G\langle g \rangle = G.
  6. Thus GG is cyclic, with generator gg.

Idea

A prime has only two divisors. An element that is not the identity cannot have order one, so its order is the full order of the group.

Reading path

To understand this page from the ground up, read these 47 pages in order. The foundations come first. The path takes the proofs Proof that the order of an element divides the order of the group, Proof of Lagrange's theorem.

  1. Divisibility definition
  2. Element definition
  3. Group concept
  4. Left coset concept
  5. Proposition concept
  6. Prime number definition
  7. Set definition
  8. Laws of exponents in a group statement
  9. Order of a group definition
  10. Subgroup definition
  11. Conjunction definition
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Empty set object
  16. Family of sets definition
  17. Ordered pair definition
  18. Order of an element definition
  19. Intersection definition
  20. Union definition
  21. Cyclic subgroup generated by an element definition
  22. Left coset definition
  23. Right coset definition
  24. Set-builder notation concept
  25. Existential quantifier definition
  26. Universal quantifier definition
  27. Cartesian product definition
  28. Disjoint sets definition
  29. g to the n equals 1 iff the order divides n statement
  30. Subset definition
  31. Union of a family of sets definition
  32. Cyclic group definition
  33. Cosets have the same cardinality as the subgroup statement
  34. Binary relation definition
  35. Partition of a set definition
  36. Order of a cyclic subgroup equals the order of its generator statement
  37. Reflexive relation definition
  38. Symmetric relation definition
  39. Transitive relation definition
  40. Equivalence relation definition
  41. Equivalence class definition
  42. Equivalence classes form a partition statement
  43. Proof of Lagrange's theorem proof
  44. Lagrange's theorem statement
  45. Proof that the order of an element divides the order of the group proof
  46. Order of an element divides the order of the group statement
  47. Proof that groups of prime order are cyclic proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory), Applications
About this page
TagG5PTXQ0QK3
URIm2://G5PTXQ0QK3
Kindproof
WritingW0 stub
MSC20D60
ProvesGroups of prime order are cyclic
UsesOrder of an element divides the order of the group, Prime number, Cyclic subgroup generated by an element, Cyclic group
MethodsDirect proof, Proof by cases

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