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

Proofs: 1

Explanations: 1
Lemmas: 2 3
Proofs: 4 5 6 7 8
Theorems: 9

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