Past exam of the options II course of the University of Oxford / 2013 / B2 / b / Solution

On the set of left cosets $G/H$, define $g\cdot (xH)=(gx)H$. Given $xH$ and $yH$, the element $y{x}^{-1}$ sends the first to the second. Thus the coset action is transitive. In particular the orbit of $H$ is the entire coset space, and its stabilizer is $H$.