Math 2.0

example

The cosets of 3Z in the integers

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}.

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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
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Sources

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

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