Math 2.0

Group theory and generalizations Example

The even integers form a subgroup of the integers

Builds on IntegersSubgroupNormal subgroup

Example

Let G=(ℤ,+)G = (\mathbb{Z}, +) be the integers with addition, and let 2ℤ2\mathbb{Z} be the set of even integers. The identity 00 is even. The sum of two even integers is even, and the negative of an even integer is even. Thus 2ℤ2\mathbb{Z} is a subgroup of GG. GG is abelian, so g+2ℤ=2ℤ+gg + 2\mathbb{Z} = 2\mathbb{Z} + g for each gg. Thus 2ℤ2\mathbb{Z} is normal in GG.

Remarks

In the same way, nℤn\mathbb{Z} is a subgroup of ℤ\mathbb{Z} for each n≥0n \ge 0.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptIntegers, object, no evidence levelIntegersobjectConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionThe even integers form a subgroup of the integers, example, no evidence levelThe even integers for…example
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  1. Group concept
  2. Integers object
  3. Conjugate elements definition
  4. Subgroup definition
  5. Conjugate subgroup definition
  6. Normal subgroup definition
  7. The even integers form a subgroup of the integers example

Further reading

  1. Web https://en.wikipedia.org/wiki/Subgroup, Other examples
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Kindexample
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MSC20A05
Instance ofSubgroup, Normal subgroup
UsesIntegers, Subgroup, Normal subgroup

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