Math 2.0

Group theory and generalizations Example

Infinite cyclic group

Builds on Cyclic groupAdditive group of integers

Example

The additive group of integers (ℤ,+)(\mathbb{Z}, +) is an infinite cyclic group. The element 11 generates it, because each integer is a sum of copies of 11 or of −1-1. Its only generators are 11 and −1-1.

Remarks

If G=⟨g⟩G = \langle g \rangle is an infinite cyclic group, then the powers gkg^k for k∈ℤk \in \mathbb{Z} are all distinct. Then k↦gkk \mapsto g^k is an isomorphism from ℤ\mathbb{Z} to GG, thus each infinite cyclic group is isomorphic to (ℤ,+)(\mathbb{Z}, +). The group of the translations of the plane by integer multiples of a fixed vector is an infinite cyclic group.

Reading path

To understand this page from the ground up, read these 10 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 groupdefinitionInfinite cyclic group, example, no evidence levelInfinite cyclic groupexample
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. Infinite cyclic group example

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclic_group, Integer and modular addition
  2. Web https://en.wikipedia.org/wiki/Cyclic_group, Definition and notation
About this page
Tag02B1T50JD1
URIm2://02B1T50JD1
Kindexample
WritingW2 usable
MSC20A05
Instance ofCyclic group
UsesCyclic group, Additive group of integers

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