Group theory and generalizations Definition

Group homomorphism

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Definition

Let $G$ and $H$ be groups. A group homomorphism from $G$ to $H$ is a map $\varphi \colon G \to H$ such that $\varphi(ab) = \varphi(a) \varphi(b)$ for all $a, b \in G$ . Here the product $ab$ is taken in $G$ and the product $\varphi(a) \varphi(b)$ is taken in $H$ .

Remarks

The map $n \mapsto 2^n$ is a homomorphism from $(\mathbb{Z}, +)$ to $(\mathbb{Q}^\times, \cdot)$ . The map $x \mapsto e^x$ is a homomorphism from $(\mathbb{R}, +)$ to the positive real numbers with multiplication. For each pair of groups, the map that sends each element of $G$ to the identity of $H$ is a homomorphism, the trivial homomorphism. The composite of two homomorphisms is a homomorphism.

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Group concept
Group homomorphism definition

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Tag	`0HV9Y7WG5R`
URI	`m2://0HV9Y7WG5R`
Aliases	`homomorphism-of-groups`
Kind	definition
Writing	W2 usable
MSC	20A05
Defines	Group homomorphism
Uses	Group

Group homomorphism

Definition

Remarks

Reading path

Where this leads

Further reading

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