Group theory and generalizations Definition

Kernel of a homomorphism

Definition

Let $G$ and $H$ be groups, let $e_H$ be the identity element of $H$ , and let $\varphi \colon G \to H$ be a group homomorphism. The kernel of $\varphi$ is the set $\ker \varphi = \{g \in G : \varphi(g) = e_H\}$ .

Remarks

The kernel is a normal subgroup of $G$ . The kernel of the map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ , $k \mapsto k + n\mathbb{Z}$ , is $n\mathbb{Z}$ . The kernel of the determinant map from $\mathrm{GL}_n(\mathbb{R})$ to $\mathbb{R}^\times$ is the special linear group $\mathrm{SL}_n(\mathbb{R})$ .

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Group concept
Group homomorphism definition
Kernel of a homomorphism definition

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Tag	`0Z424HK8CQ`
URI	`m2://0Z424HK8CQ`
Aliases	`kernel-of-a-group-homomorphism`
Kind	definition
Writing	W1 stated
MSC	20A05
Defines	Kernel of a homomorphism
Uses	Group homomorphism

Kernel of a homomorphism

Definition

Remarks

Reading path

Where this leads

Further reading

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