# The conjugate closure of a subset is a kernel of a permutation representation associated to a group action

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.

• 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 $N$. – Derek Holt Sep 1 '15 at 14:50
• @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. – bfhaha Sep 1 '15 at 18:56
• This is just Cayley's Theorem applied to $G/N$. – Derek Holt Sep 2 '15 at 14:10

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 $$N$$.