Math 2.0

Group theory and generalizations Theorem

Product of cyclic groups is cyclic iff the orders are coprime

Builds on Cyclic groupCyclic groups of the same order are isomorphicOrder of an element

Statement

Let GG and HH be finite cyclic groups of orders mm and nn. Let G×HG \times H be the group of pairs (g,h)(g, h) with g∈Gg \in G and h∈Hh \in H, with the operation (g,h)(g′,h′)=(gg′,hh′)(g, h)(g', h') = (gg', hh'). Then G×HG \times H is cyclic if and only if gcd⁡(m,n)=1\gcd(m, n) = 1.

Remarks

When gcd⁡(m,n)=1\gcd(m, n) = 1, the group ℤ/mnℤ\mathbb{Z}/mn\mathbb{Z} is isomorphic to ℤ/mℤ×ℤ/nℤ\mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}, and this is a form of the Chinese remainder theorem. When gcd⁡(m,n)>1\gcd(m, n) > 1, each element of G×HG \times H has order at most lcm⁡(m,n)<mn\operatorname{lcm}(m, n) < mn. For example, ℤ/12ℤ\mathbb{Z}/12\mathbb{Z} is isomorphic to ℤ/3ℤ×ℤ/4ℤ\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}, but ℤ/2ℤ×ℤ/6ℤ\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z} is not cyclic.

Reading path

To understand this page from the ground up, read these 11 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 …statementProduct of cyclic groups is cyclic iff the orders are coprime, statement, no evidence levelProduct of cyclic gro…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
  11. Product of cyclic groups is cyclic iff the orders are coprime statement

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Additional properties
About this page
Tag08FDFEF8D1
URIm2://08FDFEF8D1
Kindstatement, theorem
WritingW0 stub
MSC20K01
StatusThis copy of the site has no build data. Run m2 status 08FDFEF8D1.
UsesCyclic group, Cyclic groups of the same order are isomorphic, Order of an element

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