let $G$ be a finite group and $H$ any subgroup. $\psi_{H}$ be the left action of $G$ on $G/H$. It was asked to prove that the action is transitive and the kernel of $\psi_{H}$ is the 'largest normal subgroup'. It was easy to see that it is transitive.
What does this 'largest normal subgroup' mean? Is it some thing anologous to maximal ideal definition? Or does it mean the normal subgroup with greatest cardinality? (note that $G$ need not be finite). Thanks
Edit: By $G/H$ I mean the set of left cosets.