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## Lemma: Kernel and Image of Group Homomorphism

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

1. $$\ker(f)=\{e_G\}$$ $$\Longleftrightarrow f$$ is injective.
2. $$\operatorname{im}(f)=H$$ $$\Longleftrightarrow f$$ is surjective.

| | | | | created: 2014-08-28 22:42:23 | modified: 2020-06-26 17:29:10 | by: bookofproofs | references: [696]

## 1.Proof: (related to "Kernel and Image of Group Homomorphism")

### Bibliography (further reading)

[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013