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Let $G \xrightarrow{\phi} H$ be a group homomorphism.

(a) Prove that the kernel of $\phi$ is a normal subgroup of G.

(b) Prove that the image of $\phi$ is a subgroup of H.

(c) Prove that there is a group isomorphism between the quotient group $G / \ker\phi$ and the image of $\phi$.

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  • $\begingroup$ Hints: a: Try the definition??? $\endgroup$ – RougeSegwayUser Apr 22 '13 at 5:19
  • $\begingroup$ b) Again, verify the definition. c) map g->gK $\endgroup$ – RougeSegwayUser Apr 22 '13 at 5:19
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For $a)$

I write $f$ for $\phi$. We need to prove that: $\forall g\in G, \forall h\in\ker(f):ghg^{-1}\in \ker(f)$

Let $g\in G$ and let $h\in\ker(f)$. Then \begin{align*} f(ghg^{-1})&=f(g)f(h)f(g)^{-1} &&f \text{ is an homomorphism}\\ &=f(g)e_2f(g)^{-1} &&h\in\ker(f)\\ &=e_2 &&f(g)f(g)^{-1}=e_2 \end{align*}

By defintion of the kernel, then $ghg^{-1} \in \ker(f)$.

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These are in the text by Fraleigh, A first course in Abstract algebra 7ed, for 1 see Theorem 13.15 and Cor. 13.20. For 2 see T. 13.12 and for 3 see T. 14.1.

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