If $G$ act on the set of all right cosets of a subgroup $K$ then I have the following questions:-
- What is the stabilizer of an element $Kx$.
- What is the kernel of the Action.
- What is the orbit of $Kx$.
My attempt ( I know its short):-
- $Stb_G(Kx)=\{g\in G : Kxg=Kx\}$.
- $Ker = \{g\in G : Kxg=Kx \, \, \mbox{for all} \, Kx \}$ also I know the Ker is intersection of all stabilizer but since the stabilizer is not nice not sure if useful to use it.
- Okay the action is transitive, but not sure why.
Best Answer
For 1, it follows from what you wrote that $xg=k'x$ for some $k'\in K$, so $g=x^{-1}k'x$ and the answer is that the the stabilizer is $x^{-1}Kx$.
The kernel is, as you say, the intersection of all $x^{-1}K x$ for all $x\in G$. I'm not sure if it has a name: in case of commutative groups it is all of $K$, in some cases it can even be trivial. (It is always a normal subgroup of $G$.)
For transitivity, note that, in a group, you can convert an element $kx$ to any other element by suitable right multiplication.