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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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    @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.