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Suppose we have $\varphi,\psi : H \to \operatorname{Aut}(N)$, so that $\varphi = \operatorname{Ad}_g \circ \psi$ where $\operatorname{Ad}_g:N\to N$ is given by $n\mapsto gng^{-1}$ for some $g \in N$.

Prove that $N \rtimes_{\varphi} H \cong N \rtimes_{\psi} H $.

I understand it's not the same conditions as here: When are two semidirect products isomorphic?

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  • $\begingroup$ Find an isomorphism between the two groups. Remember that an element of a semidirect product can be written as an ordered pair. So where should $(n,h)$ map to? $\endgroup$ – ChocolateAndCheese May 21 '14 at 16:50
  • $\begingroup$ So, I think it is not true, but i can't find example. $\endgroup$ – Alex-omsk May 22 '14 at 17:34

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