Proof
(related to Lemma: Group Homomorphisms and Normal Subgroups)
- By hypothesis, \(f:(G,\ast)\mapsto (H,\cdot)\) is a group homomorphism.
- Due to the corresponding lemma, the kernel \(\ker(f)\) is a subgroup of \(G\).
- It remains to be shown that it is a normal subgroup of $G.$
- Let $g\in G$ and \(h\in \ker(f)\), i.e. $f(h)=e_H.$
- It follows from the properties of group homomorphism that $$\begin{array}{rcl}f(g\ast h\ast g^{-1})&=&f(g)\cdot f(h)\cdot f(g^{-1})\\&=&f(g)\cdot e_H\cdot (f(g))^{-1}\\&=&f(g)\cdot f(g)^{-1}\\&=&e_H.\end{array}$$
- Thus, $g\ast h\ast g^{-1}\in \ker(f)$ for any $g\in G$ and any $h\in \ker(f).$
- Therefore, $\ker(f)$ is a normal subgroup of \(G\), i.e. $\ker(f)\unlhd G.$
∎
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References
Bibliography
- Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013
