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Let $G$ a group, $H \le G$ and $A= G/H $. Then there exists an action $\phi: G\rightarrow S_A$ such that the kernel is the maximum subgroup normalized by $G$ and contained in $H$.

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Well, what is the simplest operation you can imagine that takes an element of $G$ and a coset in $G/H$ to produce another coset? –  Hagen von Eitzen Sep 4 '12 at 6:06
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up vote 1 down vote accepted

the homomorfism is well defined for $g\in G$: $\phi_g (aH)=gaH $ for $ aH\in G/H$.
then

$$\begin{align*} \ker\phi_g&= \{g\in G:\phi_g(aH)= aH\ \forall a\in G \}\\ &=\{g\in G: gaH=aH\ \forall a\in G\}\\ &=\{g\in G:a^{-1}ga \in H\ \forall a\in G\}\\ &=\bigcap\limits_{x\in G}aHa^{-1} \end{align*}$$

next any normal subgroup is in $\ker\phi_g$

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You still have to show that $\bigcap_{x\in G}aHa^{-1}$ is normal in $G$, and that if $N\le H$ is normal in $G$, then $N\subseteq\bigcap_{x\in G}aHa^{-1}$. –  Brian M. Scott Sep 4 '12 at 7:56
    
but a theorem said that $ker\phi_g$ always is normal at group, and containment results from $N\le G$ –  Camilo Acevedo. Sep 4 '12 at 9:57
    
I’ll grant that it’s pretty obvious that $\ker\varphi_t$ is normal in $G$, but you have to do a little work to justify the claim that if $N\le H$ is normal in $G$, then $N\subseteq_{x\in G}aHa^{-1}$. –  Brian M. Scott Sep 4 '12 at 19:31
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