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!
Best Answer
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 $\;H\;$ .