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Possible Duplicate:
Proof of first isomorphism theorem of group

Let $G_1, G_2$ be groups. If $f: G_1 \rightarrow G_2$ is a group homomorphism with $K = \ker(f)$, then $G_1 / K$ is isomorphic to $f(G_1)$.

This was a theorem in the book that was left unproven and I'm really curious as to how you would go about it.

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    @Student Yes, your first time through things may not be routine and that is okay, even if others act like it should be.2012-09-06

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You can get this theorem from any starting Algebra book, you can refer Herstein. But why not try yourself?

First prove that $\ker (f)$ is subgroup in $G_1$ (in fact it's a normal subgroup). Then

Define $h: G_1/\ker(f) = G_2$ by $h(g_1 + (\ker f)) = f(g_1)$, and check if this is isomorphism or not.

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    Thanks rschwieb for editing, I will start learning Latex.2012-09-06