Math 2.0

Group theory and generalizations Corollary

g to the order of the group is the identity

Builds on Order of an element divides the order of the groupg to the n equals 1 iff the order divides n

Statement

Let GG be a finite group with identity element ee, and let g∈Gg \in G. Then g∣G∣=eg^{|G|} = e.

Remarks

The order of gg divides ∣G∣|G| by this corollary, and a power is ee when the order divides the exponent. For the group (ℤ/pℤ)×(\mathbb{Z}/p\mathbb{Z})^\times of order p−1p - 1, with pp prime, this gives Fermat's little theorem ap−1≡1(modp)a^{p-1} \equiv 1 \pmod p for p∤ap \nmid a. For the group (ℤ/nℤ)×(\mathbb{Z}/n\mathbb{Z})^\times of order φ(n)\varphi(n), it gives Euler's theorem.

Reading path

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

Further reading

  1. Web https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory), Applications
About this page
Tag0ZEPNS9BXK
URIm2://0ZEPNS9BXK
Aliasespower-of-the-group-order-is-identity
Kindstatement, corollary
WritingW0 stub
MSC20D60
StatusThis copy of the site has no build data. Run m2 status 0ZEPNS9BXK.
UsesOrder of an element divides the order of the group, g to the n equals 1 iff the order divides n

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