Math 2.0

Group theory and generalizations Proposition

Subgroups of the integers

Builds on Subgroups of cyclic groups are cyclicAdditive group of integers

Statement

Let HH be a subgroup of the additive group of integers (ℤ,+)(\mathbb{Z}, +). Then there is exactly one integer n≥0n \geq 0 with H=nℤ={nk:k∈ℤ}H = n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}.

Remarks

When H≠{0}H \neq \{0\}, the integer nn is the smallest positive element of HH. For m,n≥0m, n \geq 0, mℤ⊆nℤm\mathbb{Z} \subseteq n\mathbb{Z} holds exactly when nn divides mm. A consequence is that, for integers aa and bb, the subgroup aℤ+bℤa\mathbb{Z} + b\mathbb{Z} is dℤd\mathbb{Z} with d=gcd⁡(a,b)d = \gcd(a, b), which gives Bezout's identity.

Reading path

To understand this page from the ground up, read these 11 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 levelCommutativitydefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionAdditive group of integers, example, no evidence levelAdditive group of int…exampleCyclic group, definition, no evidence levelCyclic groupdefinitionSubgroups of cyclic groups are cyclic, statement, no evidence levelSubgroups of cyclic g…statementSubgroups of the integers, statement, no evidence levelSubgroups of the inte…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.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Laws of exponents in a group statement
  5. Subgroup definition
  6. Abelian group definition
  7. Cyclic subgroup generated by an element definition
  8. Additive group of integers example
  9. Cyclic group definition
  10. Subgroups of cyclic groups are cyclic statement
  11. Subgroups of the integers statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Subgroups
About this page
Tag0SRATCFF1W
URIm2://0SRATCFF1W
Kindstatement, proposition
WritingW0 stub
MSC20K27
StatusThis copy of the site has no build data. Run m2 status 0SRATCFF1W.
UsesSubgroups of cyclic groups are cyclic, Additive group of integers

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