Math 2.0

Group theory and generalizations Example

Doubling on the integers is not an isomorphism

Builds on IntegersGroup isomorphismInjective functionSurjective function

Example

Let G=(ℤ,+)G = (\mathbb{Z}, +) be the integers with addition. Define d:G→Gd : G \to G by d(u)=2ud(u) = 2u. Then d(u+v)=2u+2v=d(u)+d(v)d(u + v) = 2u + 2v = d(u) + d(v), so dd is a homomorphism. If 2u=02u = 0, then u=0u = 0. Thus the kernel is {0}\{0\}, and dd is injective. The image of dd is the set of even integers. Thus 11 is not in the image, and dd is not surjective. Thus dd is not an isomorphism.

Remarks

The image 2ℤ2\mathbb{Z} is isomorphic to ℤ\mathbb{Z}. Thus dd is an isomorphism from ℤ\mathbb{Z} onto 2ℤ2\mathbb{Z}, but not onto ℤ\mathbb{Z}.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionGroup, concept, no evidence levelGroupconceptIntegers, object, no evidence levelIntegersobjectProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionDoubling on the integers is not an isomorphism, example, no evidence levelDoubling on the integ…example
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. Element definition
  2. Group concept
  3. Integers object
  4. Proposition concept
  5. Set definition
  6. Group homomorphism definition
  7. Implication definition
  8. Predicate definition
  9. Ordered pair definition
  10. Group isomorphism definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Codomain definition
  19. Injective function definition
  20. Surjective function definition
  21. Doubling on the integers is not an isomorphism example

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_isomorphism, Definition and notation
  2. Web https://en.wikipedia.org/wiki/Group_homomorphism, Types
About this page
TagPVR5EX71PD
URIm2://PVR5EX71PD
Kindexample
WritingW0 stub
MSC20A05
Instance ofGroup homomorphism
Not an instance ofGroup isomorphism (note: the image is the even integers, so the map is not onto.)
UsesIntegers, Group isomorphism, Injective function, Surjective function

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