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Question 1

Let $G$ be a group and $H\unlhd G$ and $K\leq G$ such that $H\cap K = 1$ and $HK=G$

Let $\varphi:K\rightarrow Aut(H)$ be a homomorphism such that $HK\cong H\rtimes_\varphi K$.

Then is $\varphi$ trivial if $K\unlhd G$? And is $K\unlhd G$ if $\varphi$ is trivial?

Question 2

Let $H,K$ be groups and $\varphi_1,\varphi_2:K\rightarrow Aut(H)$ be homomorphisms such that $H\rtimes_{\varphi_1} K\cong H\rtimes_{\varphi_2} K$.

Then, is there a relation between $\varphi_1$ and $\varphi_2$?

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In general, if $\phi=\varphi\pmod{\operatorname{Inn}(H)}$ then $H\rtimes_{\phi}K\cong H\rtimes_{\varphi}K$, where $\operatorname{Inn}(H)$ denotes the inner automorphisms of $H$ (automorphisms of the form $\gamma_g: h\mapsto g^{-1}hg$).

This then answers you first part of the first question. Can you see why? (The answer to the second part is "yes", and the proof is simple - think about direct products, or do it explicitly).

The answer to your second question is more subtle. Under certain circumstances (for example, $H$ does not map onto $\mathbb Z$ and $K$ is infinite cyclic) the above conditions are necessary and sufficient. For a counter-example, look up the classification of groups of order $pq$ ($p, q$ prime. There I no more than two groups of order $pq$, but there are more than two relevant maps (and no non-trivial inner ones!).

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    $\begingroup$ I don't get the first question part.. Would you please give me more details? $\endgroup$ – Rubertos Feb 18 '15 at 8:17
  • $\begingroup$ And I think I should edit the hypothesis as $K\unlhd HK$ rather than $K\unlhd G$ $\endgroup$ – Rubertos Feb 18 '15 at 8:20
  • $\begingroup$ For Question 2, see math.stackexchange.com/questions/527800 $\endgroup$ – Derek Holt Feb 18 '15 at 8:45
  • $\begingroup$ @Rubertos What kind of details Are you after? Do you know the definition of an inner automorphism? $\endgroup$ – user1729 Feb 18 '15 at 18:40

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