Group theory and generalizations Example

Reduction modulo n is a homomorphism

Builds on Integers Integers modulo n Kernel of a homomorphism

Example

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

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Group concept
Integers object
Integers modulo n concept
Group homomorphism definition
Kernel of a homomorphism definition
Reduction modulo n is a homomorphism example

Tag	`BV08DGBD7C`
URI	`m2://BV08DGBD7C`
Kind	example
Writing	W0 stub
MSC	20A05
Instance of	Group homomorphism
Illustrates	First isomorphism theorem
Uses	Integers, Integers modulo n, Kernel of a homomorphism

Reduction modulo n is a homomorphism

Example

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