Let $H$ be a subgroup of $G$, then $G$ acts on the left cosets $Cos_H$ of $H$ by left multiplication. Show that $ker(\phi)$ of the homomorphism $\phi : G \to Sym(Cos_H)$ is the largest subgroup in $H$ that is normal in $G$.
Can anyone help me with this one? Is $Cos_H$ the set of all left cosets? So $\phi$ must be a function that takes an element in $G$ and throws it on a left coset? and is $Sym(Cos_H)$ the set of al permutations with elements in $Cos_H$?