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I would like some help on the following problem from anyone who would like to help.

Let $f: H \to G$ be a group homomorphism. For $h \in H$, define $\rho(h) = \phi_{f(h)} \in Aut(G)$.

The situation being as described, prove that the semi-direct product $G\rtimes_{\rho} H$ is isomorphic to the direct product $G \times H$.

Help will be greatly appreciated!

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    why don't you try $T(g, h) = (gf(h), h)$?2012-12-03

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