Kernel of action of G on set of cosets of H in G is contained in H

Question

Let $G$ be a group and $H$ a subgroup of G. Also, let $X$ be the set of left cosets, $xH$, of H in G. Define an action of $G$ on $X$ by $g \cdot xH = gxH$ for $g,x \in G$.

I have shown that the kernel, $K$, where $K=\bigcap_{xH \in X}xHx^{-1}$ and $Stab_{G}(xH) = xHx^{-1}$ ($Stab$ is the stabiliser of $xH$) of the action is a normal subgroup of G, but I don't know how to show that $K \subseteq H$. I've considered taking an element of the kernel and tried to show it is an element of $H$ but haven't got anywhere. I've also looked at the kernel in terms a homomorphism but I feel like I'm getting nowhere.

Any help would be appreciated!

score 2 · Answer 1 · answered Feb 18 '16 at 23:29

Suppose $\;k\in K\;$ , then for all $\;g\in G\;$ we have that $\;k\cdot gH=gH\;$ , and in particular $\;kH=H\iff k\in H\;$ .

$\;K\;$ is called the core of the subgroup $\;H\;$ , and it is the maximal normal subgroup of $\;G\;$ which is contained in $\;H\;$ .

score 1 · Accepted Answer · answered Feb 18 '16 at 23:29

1

Take $x=e$. Then $xHx^{-1}=H$. The intersection of some collection of sets is a subset of every set in the collection. Hence in this case the intersection is a subset of $H$.

answered Feb 18 '16 at 23:29

Matt Samuel

59,287

Could you explain why we can take $x = e$? I mean, couldn't we choose $x$ to be something else, aside from $e$? – Joe Feb 18 '16 at 23:32
@Joseph sure. The intersection is contained in $xHx^{-1}$ for all $x$. In particular it's true when $x=e$, which is the question you asked. – Matt Samuel Feb 18 '16 at 23:34
Of course, thanks for the rapid response @Matt Samuel. – Joe Feb 18 '16 at 23:36
@Joseph no problem. – Matt Samuel Feb 18 '16 at 23:41

Kernel of action of G on set of cosets of H in G is contained in H

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