Math 2.0

Group theory and generalizations Theorem

Universal property of the quotient group

Builds on Quotient groupGroup homomorphismKernel of a homomorphism

Statement

Let GG and HH be groups, let NN be a normal subgroup of GG, and let π ⁣:G→G/N\pi \colon G \to G/N be the projection π(g)=gN\pi(g) = gN to the quotient group. Let φ ⁣:G→H\varphi \colon G \to H be a group homomorphism with N⊆ker⁡φN \subseteq \ker \varphi. Then there is exactly one group homomorphism φˉ ⁣:G/N→H\bar{\varphi} \colon G/N \to H with φˉ∘π=φ\bar{\varphi} \circ \pi = \varphi, and it is given by φˉ(gN)=φ(g)\bar{\varphi}(gN) = \varphi(g).

Remarks

The map φˉ\bar{\varphi} is well defined because φ\varphi is constant on each coset of NN. The image of φˉ\bar{\varphi} is the image of φ\varphi, and the kernel of φˉ\bar{\varphi} is (ker⁡φ)/N(\ker \varphi)/N. With N=ker⁡φN = \ker \varphi, this gives the first isomorphism theorem.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionUniversal property of the quotient group, statement, no evidence levelUniversal property 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.
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  1. Group concept
  2. Conjugate elements definition
  3. Group homomorphism definition
  4. Subgroup definition
  5. Conjugate subgroup definition
  6. Kernel of a homomorphism definition
  7. Left coset definition
  8. Right coset definition
  9. Normal subgroup definition
  10. Normality via left and right cosets statement
  11. Quotient group definition
  12. Universal property of the quotient group statement

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_group, Properties
About this page
Tag0GJ83E78A5
URIm2://0GJ83E78A5
Aliasesfactorization-through-the-quotient
Kindstatement, theorem
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0GJ83E78A5.
UsesQuotient group, Group homomorphism, Kernel of a homomorphism

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