Math 2.0

Group theory and generalizations Example

Integers modulo n as a quotient group

Builds on Quotient groupAdditive group of integersSubgroups of the integers

Example

Let nn be a positive integer. The subgroup nℤn\mathbb{Z} of the additive group of integers is normal, because ℤ\mathbb{Z} is abelian. The quotient group ℤ/nℤ\mathbb{Z}/n\mathbb{Z} has the nn cosets 0+nℤ,1+nℤ,…,(n−1)+nℤ0 + n\mathbb{Z}, 1 + n\mathbb{Z}, \ldots, (n-1) + n\mathbb{Z}. The operation is (a+nℤ)+(b+nℤ)=(a+b)+nℤ(a + n\mathbb{Z}) + (b + n\mathbb{Z}) = (a + b) + n\mathbb{Z}, which is addition modulo nn.

Remarks

The coset a+nℤa + n\mathbb{Z} is the set of integers with the same remainder as aa on division by nn. Thus ℤ/nℤ\mathbb{Z}/n\mathbb{Z} is the cyclic group of order nn of clock arithmetic. For n=0n = 0, the quotient ℤ/0ℤ\mathbb{Z}/0\mathbb{Z} is isomorphic to ℤ\mathbb{Z}.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionAdditive group of integers, example, no evidence levelAdditive group of int…exampleCyclic group, definition, no evidence levelCyclic groupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionSubgroups of cyclic groups are cyclic, statement, no evidence levelSubgroups of cyclic g…statementNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSubgroups of the integers, statement, no evidence levelSubgroups of the inte…statementQuotient group, definition, no evidence levelQuotient groupdefinitionIntegers modulo n as a quotient group, example, no evidence levelIntegers modulo n as …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. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Conjugate elements definition
  5. Laws of exponents in a group statement
  6. Subgroup definition
  7. Abelian group definition
  8. Conjugate subgroup definition
  9. Cyclic subgroup generated by an element definition
  10. Left coset definition
  11. Right coset definition
  12. Additive group of integers example
  13. Cyclic group definition
  14. Normal subgroup definition
  15. Subgroups of cyclic groups are cyclic statement
  16. Normality via left and right cosets statement
  17. Subgroups of the integers statement
  18. Quotient group definition
  19. Integers modulo n as a quotient group example

Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_group, Integer modular arithmetic
  2. Web https://en.wikipedia.org/wiki/Quotient_group, Remainders of integer division
About this page
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Kindexample
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MSC20K01
Instance ofQuotient group
UsesQuotient group, Additive group of integers, Subgroups of the integers

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