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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")

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Bibliography (further reading)

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