Math 2.0

Group theory and generalizations Definition

Presentation of a group

Builds on Free groupNormal subgroupQuotient groupGroup isomorphism

Definition

Let SS be a set, let FSF_S be the free group on SS, and let R⊆FSR \subseteq F_S. Let NN be the smallest normal subgroup of FSF_S that contains RR, the normal closure of RR. The group with presentation ⟨S∣R⟩\langle S \mid R \rangle is the quotient group FS/NF_S / N. A group GG has the presentation ⟨S∣R⟩\langle S \mid R \rangle when GG is isomorphic to FS/NF_S / N.

Remarks

The elements of SS are the generators and the elements of RR are the relators; a relation x=yx = y stands for the relator y−1xy^{-1} x. The cyclic group of order nn has the presentation ⟨a∣an⟩\langle a \mid a^n \rangle, and the dihedral group of order 1616 has the presentation ⟨r,f∣r8,f2,(rf)2⟩\langle r, f \mid r^8, f^2, (r f)^2 \rangle. Each group has a presentation, and each finite group has a presentation with SS and RR finite. A group is finitely presented when it has a presentation with SS and RR finite.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

Show the path as a graph
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 isomorphismdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionFree group, definition, no evidence levelFree groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormality 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…definition
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. Left coset definition
  9. Right coset definition
  10. Free group definition
  11. Normal subgroup definition
  12. Normality via left and right cosets statement
  13. Quotient group definition
  14. Presentation of a group definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Presentation_of_a_group, Definition
  2. Web https://en.wikipedia.org/wiki/Presentation_of_a_group, Finitely presented groups
About this page
Tag0QZCY9PE3Q
URIm2://0QZCY9PE3Q
Aliasesgenerators-and-relations
Kinddefinition
WritingW1 stated
MSC20F05
DefinesPresentation of a group
UsesFree group, Normal subgroup, Quotient group, Group isomorphism

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by