Math 2.0

Group theory and generalizations Example

The cosets of 3Z in the integers

Builds on IntegersLeft cosetIntegers modulo n

Example

Let G=(ℤ,+)G = (\mathbb{Z}, +) be the integers with addition, and let H=3ℤH = 3\mathbb{Z} be the multiples of 33. The cosets of HH are 3ℤ3\mathbb{Z}, 1+3ℤ1 + 3\mathbb{Z} and 2+3ℤ2 + 3\mathbb{Z}. Each integer is in exactly one of them, given by its remainder after division by 33. The group is abelian, so each left coset is also a right coset.

Remarks

The three cosets are the elements of the group ℤ/3ℤ\mathbb{Z}/3\mathbb{Z}.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptIntegers, object, no evidence levelIntegersobjectIntegers modulo n, concept, no evidence levelIntegers modulo nconceptSubgroup, definition, no evidence levelSubgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionThe cosets of 3Z in the integers, example, no evidence levelThe cosets of 3Z in t…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. Group concept
  2. Integers object
  3. Integers modulo n concept
  4. Subgroup definition
  5. Left coset definition
  6. The cosets of 3Z in the integers example

Further reading

  1. Web https://en.wikipedia.org/wiki/Coset, Integers
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