Math 2.0

Group theory and generalizations Theorem

Automorphism group of a cyclic group

Builds on Automorphism groupCyclic groupOrder of a powerGroup of units of a monoid

Statement

Let nn be a positive integer and let GG be a cyclic group of order nn. For each integer aa with gcd⁡(a,n)=1\gcd(a, n) = 1, the map σa ⁣:x↦xa\sigma_a \colon x \mapsto x^a is an automorphism of GG. The map a mod n↦σaa \bmod n \mapsto \sigma_a is an isomorphism from the group of units (ℤ/nℤ)×(\mathbb{Z}/n\mathbb{Z})^\times to Aut⁡(G)\operatorname{Aut}(G).

Remarks

An automorphism of G=⟨g⟩G = \langle g \rangle is determined by the image of gg, and this image must be a generator gag^a with gcd⁡(a,n)=1\gcd(a, n) = 1. Thus ∣Aut⁡(G)∣=φ(n)|\operatorname{Aut}(G)| = \varphi(n), where φ\varphi is Euler's totient function. For the infinite cyclic group, Aut⁡(ℤ)≅{1,−1}\operatorname{Aut}(\mathbb{Z}) \cong \{1, -1\}.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptAssociativity, definition, no evidence levelAssociativitydefinitionIdentity element, definition, no evidence levelIdentity elementdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementSubgroup, definition, no evidence levelSubgroupdefinitionSemigroup, definition, no evidence levelSemigroupdefinitionInverse element, definition, no evidence levelInverse elementdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionMonoid, definition, no evidence levelMonoiddefinitionAutomorphism group, definition, no evidence levelAutomorphism groupdefinitiong to the n equals 1 iff the order divides n, statement, no evidence levelg to the n equals 1 i…statementCyclic group, definition, no evidence levelCyclic groupdefinitionGroup of units of a monoid, definition, no evidence levelGroup of units of a m…definitionOrder of a power, statement, no evidence levelOrder of a powerstatementAutomorphism group of a cyclic group, statement, no evidence levelAutomorphism group of…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Associativity definition
  4. Identity element definition
  5. Group homomorphism definition
  6. Laws of exponents in a group statement
  7. Subgroup definition
  8. Semigroup definition
  9. Inverse element definition
  10. Group isomorphism definition
  11. Order of an element definition
  12. Cyclic subgroup generated by an element definition
  13. Monoid definition
  14. Automorphism group definition
  15. g to the n equals 1 iff the order divides n statement
  16. Cyclic group definition
  17. Group of units of a monoid definition
  18. Order of a power statement
  19. Automorphism group of a cyclic group statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Automorphism_group, Examples
About this page
Tag0G4C8SJ6PV
URIm2://0G4C8SJ6PV
Kindstatement, theorem
WritingW0 stub
MSC20F28
StatusThis copy of the site has no build data. Run m2 status 0G4C8SJ6PV.
UsesAutomorphism group, Cyclic group, Order of a power, Group of units of a monoid

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