Prove that a group acts faithfully on a set if and only if the kernel of the action is the set containing only the identity.
We know that a group action is faithful precisely when the corresponding permutation representation is injective. Moreover, a group homomorphism is injective precisely when its kernel is trivial. Finally, by the previous exercise, the kernel of a group action is equal to the kernel of the corresponding permutation representation. So acts faithfully on if and only if the kernel of the action is trivial.