# semidirect product $\implies$ exact sequence

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.

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

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What are $\alpha$ and $\beta$? – Rudy the Reindeer Dec 12 '12 at 13:03
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. – Alex B. Dec 12 '12 at 13:18
im sorry that i ask a lot... wish i was as smart as you. – grendizer Dec 12 '12 at 13:59
I second Matt's comment, and would like to ask what you have tried. – Tobias Kildetoft Dec 12 '12 at 14:19
@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 $G/N\cong H$). – user1729 Dec 12 '12 at 14:32

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.