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I am doing my thesis about semidirect products mainly and wondering how to solve the following question:

Let $G$ be a semidirect product of a normal subgroup $N$ with two elements by a subgroup $H$. Show that $G$ is an internal direct product of $N$ and $H$.

(I know that a semidirect prouct of $N$ by $H$ is the direct product if and only if the homomophism $H$ to $\operatorname{Aut}(N)$ is trivial, that is $\operatorname{id}(N)$ for all $h \in H$, but I dont really know how to move forward.)

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    I think you missed some typing or you meant something else: in your third line, what does "...of a group N *who two elements by* a subgroup H" mean?2012-11-12
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    Did you mean perhaps "N is a normal subgroup *with two elements*"?2012-11-12
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    sry corected it. yeah assumption would be N being normal.2012-11-12
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    How many automorphisms does $N$ have?2012-11-12
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    the question is in humphreys a course in group theory chapter 19 question 4. I dont understand the solution fully and also would like to show it some other way if possible.2012-11-12

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