Let \((G,\ast)\) and \((H,\cdot)\) be two groups with the respective identities \(e_G\) and \(e_H\) and \(f:G\rightarrow H\) be a group homomorphism. We define:

- the
**kernel**$\operatorname{ker}(f):= \{g\in G\mid f(g)=e_H\}$, in other words, it is the fiber of $e_H$ under the group homomorphism $f$ , - the
**image**$\operatorname{im}(f):=f[G]=\{f(g)\in H\mid g\in G\},$ in other words, it is the image of $G$ under the group homomorphism $f$.

The kernel and the image of $f$ fulfill the following defining properties:

- \(\ker(f)=\{e_G\}\) \(\Longleftrightarrow f\) is injective.
- \(\ker(f)=\{e_G\}\) \(\Longleftrightarrow f\) is injective.

Proofs: 1

Explanations: 1

Lemmas: 2 3

Proofs: 4 5 6 7 8

Theorems: 9

**Kramer Jürg, von Pippich, Anna-Maria**: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013