Math 2.0

Group theory and generalizations Theorem

Cyclic groups of the same order are isomorphic

Builds on Cyclic groupOrder of a cyclic subgroup equals the order of its generatorOrder of a group

Statement

Let GG and HH be cyclic groups of the same order, finite or infinite. Then GG and HH are isomorphic: there is a bijection φ ⁣:G→H\varphi \colon G \to H with φ(ab)=φ(a)φ(b)\varphi(ab) = \varphi(a) \varphi(b) for all a,b∈Ga, b \in G.

Remarks

If gg generates GG and hh generates HH, an isomorphism is φ(gk)=hk\varphi(g^k) = h^k for k∈ℤk \in \mathbb{Z}. This map is well defined because ga=gbg^a = g^b exactly when ha=hbh^a = h^b, by the order of a cyclic subgroup. Thus each cyclic group of order nn is isomorphic to ℤ/nℤ\mathbb{Z}/n\mathbb{Z}, and each infinite cyclic group is isomorphic to ℤ\mathbb{Z}.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementOrder of a group, definition, no evidence levelOrder of a groupdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitiong 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 groupdefinitionOrder of a cyclic subgroup equals the order of its generator, statement, no evidence levelOrder of a cyclic sub…statementCyclic groups of the same order are isomorphic, statement, no evidence levelCyclic groups of the …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. Group concept
  2. Laws of exponents in a group statement
  3. Order of a group definition
  4. Subgroup definition
  5. Order of an element definition
  6. Cyclic subgroup generated by an element definition
  7. g to the n equals 1 iff the order divides n statement
  8. Cyclic group definition
  9. Order of a cyclic subgroup equals the order of its generator statement
  10. Cyclic groups of the same order are isomorphic statement

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Definition and notation
  2. Web https://en.wikipedia.org/wiki/Cyclic_group, Integer and modular addition
About this page
Tag0736AXH2ME
URIm2://0736AXH2ME
Kindstatement, theorem
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0736AXH2ME.
UsesCyclic group, Order of a cyclic subgroup equals the order of its generator, Order of a group

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