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Let be $V\leq \mathrm{Aut}\left( G\right)$, $N\vartriangleleft G$ and $V$-invariant. Consider the semidirect product $G\rtimes V$.

Let $X$ be the preimage of $C_{G/N}\left( V\right) =\left\{ gN~;~g^{v}N=gN,\forall v\in V\right\} $ in $G$. Is it true that $X$ normalizes $NV$?

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    Are you familiar with the process of "accepting" answers to your questions? Look at [this link](http://meta.math.stackexchange.com/q/3286/742) as a first step.2012-05-21

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Let $g\in X$. If $v\in V$, then we know that $vgv^{-1}N = gN$, so in particular there exists $n\in N$ such that $vgv^{-1}=gn$. Therefore, $vgv^{-1}g^{-1}=gng^{-1} = n'$ for some $n'\in N$ (since $N\triangleleft G$), so $gv^{-1}g^{-1} = v^{-1}n' = (v^{-1}n'v)v^{-1}=n''v^{-1}$ for some $n''\in N$.

Therefore, for every $n\in N$ and $v\in V$ we have $g(nv^{-1})g^{-1} = \Bigl(gng^{-1}\Bigr)\Bigl( gv^{-1}g^{-1}\Bigr) = n'''n''v^{-1}\in NV.$

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    I understood! Thank you!2012-05-21