Math 2.0

Group theory and generalizations Example

The odd integers are not a subgroup of the integers

Builds on IntegersSubgroup

Example

Let G=(ℤ,+)G = (\mathbb{Z}, +) be the integers with addition, and let OO be the set of odd integers. The identity 00 is not odd, so 0∉O0 \notin O. Also, 1∈O1 \in O and 1+1=2∉O1 + 1 = 2 \notin O. Thus OO is not closed under addition. Thus OO is not a subgroup of GG.

Remarks

OO is a coset of the even integers. A coset g+Hg + H is a subgroup only when g∈Hg \in H.

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Group, concept, no evidence levelGroupconceptIntegers, object, no evidence levelIntegersobjectSubgroup, definition, no evidence levelSubgroupdefinitionThe odd integers are not a subgroup of the integers, example, no evidence levelThe odd integers are …example
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Further reading

  1. Web https://en.wikipedia.org/wiki/Subgroup, Subgroup tests
  2. Web https://en.wikipedia.org/wiki/Quotient_group, Even and odd integers
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Not an instance ofSubgroup (note: the sum of two odd integers is even.)
UsesIntegers, Subgroup

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