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## A group action is faithful precisely when the kernel of the corresponding permutation representation is trivial

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.6

Prove that a group $G$ acts faithfully on a set $A$ if and only if the kernel of the action is the set containing only the identity.

Solution: We know that a group action is faithful precisely when the corresponding permutation representation $\varphi : G \rightarrow S_A$ is injective. Moreover, a group homomorphism is injective precisely when its kernel is trivial. Finally, by Exercise 1.7.5, the kernel of a group action is equal to the kernel of the corresponding permutation representation. So $G$ acts faithfully on $A$ if and only if the kernel of the action is trivial.