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Let $G$ be a group and $H$ be a subgroup of $G$. Let $A$ be the set of all left cosets of $H$ in $G$. We know that $G$ can acts on $A$ by $$g\cdot xH=gxH.$$

For any $g\in G$, define that $\sigma_{g}:A\to A, \sigma_g(xH)=g\cdot xH$. Then $\sigma_g$ is a permutation on $A$.

Let $S_A$ be the set of all permutation on $A$. Define the permutation representation $\theta:G\to S_A$. Then $\theta$ is a homomorphism and $$\begin{align} \ker{\theta}&=\text{core }H\\ &=\{g\in G\mid g\in aHa^{-1}\text{ for all }a\in G\}\\ &=\bigcap_{a\in G}aHa^{-1}. \end{align}$$

The dual notion of the $\text{core}$ is the normal closure (or conjugate closure) of a subset $X$ of $G$. Which is defined by $$\begin{align} \bar{X}&=\langle \{gxg^{-1}\mid g\in G, x\in X\}\rangle\\ &=\bigcap_{X\subseteq N\lhd G}N. \end{align}$$

Is there a group action whose kernel of the permutation representation is the normal closure $\bar{X}$ of $X$?

Thanks for any kind of tips.

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  • $\begingroup$ yes, just the action on the left cosets of the normal closure $\langle X^G\rangle$. For any normal subgroup $N$ of $G$, the kernel of the action on the cosets of $N$ is $N$. $\endgroup$ – Derek Holt Sep 1 '15 at 14:50
  • $\begingroup$ @DerekHolt Oh! Thanks. You are right. But is there another group action whose kernel is the normal closure. I guess that mathematician find out some group action first. Then find out its kernel. That is, the normal closure. (May be my guess is wrong.) Which book can I learn more about the theorem as you state: For any normal subgroup $N$... and related topic. $\endgroup$ – bfhaha Sep 1 '15 at 18:56
  • $\begingroup$ This is just Cayley's Theorem applied to $G/N$. $\endgroup$ – Derek Holt Sep 2 '15 at 14:10
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The following summarises the comment by @DerekHolt . . .

Consider the action of the left cosets of the normal subgroup $\langle X^G\rangle$ generated by the action of $G$ on $X$. If $N$ is a normal subgroup of $G$, then the kernel of the action of the cosets on $N$ is $N$.

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