Math 2.0

Group theory and generalizations Theorem

Von Dyck's theorem

Builds on Presentation of a groupUniversal property of the free groupUniversal property of the quotient groupGroup homomorphism

Statement

Let G=⟨S∣R⟩G = \langle S \mid R \rangle, let HH be a group, and let f ⁣:S→Hf \colon S \to H be a function. Let φ ⁣:FS→H\varphi \colon F_S \to H be the homomorphism with φ(s)=f(s)\varphi(s) = f(s) for each s∈Ss \in S. If φ(r)=e\varphi(r) = e for each r∈Rr \in R, then there is exactly one group homomorphism ψ ⁣:G→H\psi \colon G \to H with ψ(sN)=f(s)\psi(s N) = f(s) for each s∈Ss \in S.

Remarks

Here NN is the normal closure of RR in FSF_S, so G=FS/NG = F_S / N. The proof combines the universal property of the free group with the universal property of the quotient group, because N⊆ker⁡φN \subseteq \ker \varphi. If the elements f(s)f(s) generate HH, then ψ\psi is surjective, so HH is a quotient of GG. The name refers to Walther von Dyck, who gave the first systematic study of group presentations in the early 1880s.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptReduced word, definition, no evidence levelReduced worddefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionFree group, definition, no evidence levelFree groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionUniversal property of the free group, statement, no evidence levelUniversal property of…statementNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionPresentation of a group, definition, no evidence levelPresentation of a gro…definitionUniversal property of the quotient group, statement, no evidence levelUniversal property of…statementVon Dyck's theorem, statement, no evidence levelVon Dyck's theoremstatement
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. Reduced word definition
  3. Conjugate elements definition
  4. Group homomorphism definition
  5. Subgroup definition
  6. Conjugate subgroup definition
  7. Group isomorphism definition
  8. Kernel of a homomorphism definition
  9. Left coset definition
  10. Right coset definition
  11. Free group definition
  12. Normal subgroup definition
  13. Universal property of the free group statement
  14. Normality via left and right cosets statement
  15. Quotient group definition
  16. Presentation of a group definition
  17. Universal property of the quotient group statement
  18. Von Dyck's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Presentation_of_a_group, Definition
  2. Web https://en.wikipedia.org/wiki/Free_group, Universal property
About this page
Tag0QPYHHNRXC
URIm2://0QPYHHNRXC
Kindstatement, theorem
WritingW0 stub
MSC20F05
StatusThis copy of the site has no build data. Run m2 status 0QPYHHNRXC.
UsesPresentation of a group, Universal property of the free group, Universal property of the quotient group, Group homomorphism

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