Math 2.0

example

The integers modulo the even integers

Example

Let G=(Z,+)G = (\mathbb{Z}, +) be the integers with addition, and let N=2ZN = 2\mathbb{Z} be the even integers. NN is normal, because GG is abelian. NN has two cosets: NN, the even integers, and 1+N1 + N, the odd integers. The sum of two odd integers is even, so (1+N)+(1+N)=N(1 + N) + (1 + N) = N. Thus the quotient group Z/2Z\mathbb{Z}/2\mathbb{Z} has two elements, with identity NN. It is the integers modulo 22.

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To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

Group, concept, no evidence levelGroupconceptIntegers, object, no evidence levelIntegersobjectIntegers modulo n, concept, no evidence levelIntegers modulo nconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionThe integers modulo the even integers, example, no evidence levelThe integers modulo t…example
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Group concept
  2. Integers object
  3. Integers modulo n concept
  4. Conjugate elements definition
  5. Subgroup definition
  6. Conjugate subgroup definition
  7. Left coset definition
  8. Right coset definition
  9. Normal subgroup definition
  10. Normality via left and right cosets statement
  11. Quotient group definition
  12. The integers modulo the even integers example

Sources

  1. Web https://en.wikipedia.org/wiki/Quotient_group, Even and odd integers

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