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Suppose $G$ is a group and $N$ is a normal subgroup in $G$. Also suppose $G=N \rtimes H$. I need to know, is this semi-direct product reduced to the direct product if $N=Z(G)$?

My initial guess is yes! But any detailed explanation will be highly appreciated.

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  • $\begingroup$ Only of the action of $H$ on $N$ is trivial, surely? $\endgroup$ – Richard Martin Nov 9 '18 at 13:39
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    $\begingroup$ @RichardMartin But if $N = Z(G)$ (or even $N \le Z(G)$) then the action of $H$ on $N$ is trivial, so the answer to the question is yes. $\endgroup$ – Derek Holt Nov 9 '18 at 14:31

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