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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
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    This is just routine proof, if you know the definition of isomorphism then you can do it – pritam Sep 06 '12 at 07:55
  • You can find a clear proof in any group theory book of this theorem. :) – Mikasa Sep 06 '12 at 07:56
  • I can't think of a mapping that would make sense intuitively. And this is my first algebra course so nothing is 'routine' for me yet – Student Sep 06 '12 at 07:56
  • @Student: The mapping is given to you, isn't it ? – pritam Sep 06 '12 at 07:58
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    There is only one reasonable map to try: $\varphi(aK)=f(a)$. You just have to prove that $\varphi$ is well-defined, meaning that if $aK=bK$, then $f(a)=f(b)$, and that it satisfies the definition of an isomorphism. – Brian M. Scott Sep 06 '12 at 07:59
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    @pritam: Not quite, but almost. – Brian M. Scott Sep 06 '12 at 08:00
  • As others have said, this is know as the first isomorphism theorem for groups. It is one of a number of isomorphism theorems, which have analogous results for other algebraic structures. As Brian said, you need to check that the map he gave is (i) well-defined, (ii) a group homomorphism and (iii) a bijective map (hence an isomorphism). For part (i) you just want to use the definition and the fact that $f$ is a homomorphism (so what do you know about $f(a)^{-1}$ for example?); part (ii) should be fairly straightforward; for part (iii) what can you say about $\varphi$ from your knowledge of $f$? – David Ward Sep 06 '12 at 08:10
  • @BabakSorouh Any... you know, except the one OP is actually using. – GeoffDS Sep 06 '12 at 12:50
  • @Student Yes, your first time through things may not be routine and that is okay, even if others act like it should be. – GeoffDS Sep 06 '12 at 12:51

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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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