Math 2.0

example

Reduction modulo n is a homomorphism

Example

Let n≥1n \ge 1. Define φ:Z→Z/nZ\varphi : \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} by φ(u)=u mod n\varphi(u) = u \bmod n. Then φ(u+v)=φ(u)+φ(v)\varphi(u + v) = \varphi(u) + \varphi(v), so φ\varphi is a group homomorphism. The map is onto, and its kernel is nZn\mathbb{Z}, the integers that nn divides. By the first isomorphism theorem, Z/nZ\mathbb{Z}/n\mathbb{Z} as a quotient of Z\mathbb{Z} is isomorphic to the image, the integers modulo nn.

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Group, concept, no evidence levelGroupconceptIntegers, object, no evidence levelIntegersobjectIntegers modulo n, concept, no evidence levelIntegers modulo nconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionReduction modulo n is a homomorphism, example, no evidence levelReduction modulo n is…example
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  1. Group concept
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  5. Kernel of a homomorphism definition
  6. Reduction modulo n is a homomorphism example

Sources

  1. Web https://en.wikipedia.org/wiki/Group_homomorphism, Examples

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