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Hey so I´m given a semidirect product $G=NH$, where $N$ is normal in $G$ and $N\cap H=1$. I have to show that the sequence below is exact.

$$1\xrightarrow{}N\xrightarrow{\alpha}G\xrightarrow{\beta}H\xrightarrow{}1$$

I am not sure how or where to start. Please help me.

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    What are $\alpha$ and $\beta$?2012-12-12
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    Looking at the history of your questions in the past week, it looks like you are asking this site to write your thesis for you. I can't say that that particularly motivates me to give an answer.2012-12-12
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    im sorry that i ask a lot... wish i was as smart as you.2012-12-12
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    I second Matt's comment, and would like to ask what you have tried.2012-12-12
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    @MattN. Are you asking your question because you are not sure, or because you are trying to give a hint? I mean, the question can be re-phrased as `what are $\alpha$ and $\beta$?' One he knows this, he knows the answer (mod the fact that $G/N\cong H$).2012-12-12
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    @user1729 Exactly. But he clearly states that he doesn't even know where to start.2012-12-12
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    @user1729 And to answer your question: I was thinking aloud.2012-12-12

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I'm assuming that you need to pick the appropriate $\alpha$ and $\beta$ yourself. Then here is how you can do it:

Define $\alpha$ to be the natural inclusion: $\alpha(n)=n$ for every $n \in N$. Define $\beta$ by saying that $\beta(g) g^{-1} \in N$ (for every $g\in G$ there is exactly one such $\beta(g) \in H$). All you need to do now is check that $\alpha$ and $\beta$ are homomorphisms and that the sequence is exact. You know what $\alpha$ and $\beta$ are, so this is a pretty straightforward task.

If you already know this, then I've told you nothing new. If that's the case then it would really help to see what you've tried yourself.