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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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    the question is in humphre$y$s 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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You know that semidirect products $C_2\rtimes H$ require a homomorphism $\phi:H\rightarrow\text{Aut}C_2$. The product is direct if the homomorphism is trivial. Now, let's figure out the automorphisms of $C_2$. Any homomorphism $\theta$ preserves the identity, so $\theta:C_2\rightarrow C_2$ takes $\theta(1)=1$. Furthermore, any automorphism is bijective, so the only other element of $C_2$ (its generator) must be mapped to itself by $\theta$. Thus for any automorphism $\theta$, $\theta(x)=x$ for both $x\in C_2$, so $\text{Aut}C_2$ only consists of the trivial homomorphism. Thus any semidirect product must be trivial.