[Math] S3 group action faithful

Question

I'm struggling with understanding the term "faithful". I read that a group action for example $S_3$ is faithful on {1,2,3}. Does that mean $S_3$ is not faithful on {1,2,3,4} because it never changes 4? I've read that a group action is faithful when the homomorphism is injectiv ergo it has trivial kernel. But I struggle to understand that, what is the kernel of $S_3$ is on {1,2,3,4} or $S_3$ on {A,B} for example. Thanks

Answer 1

A group $G$ acts faithfully on $X$ if the identity of $G$ is the only group element that leaves all elements of $X$ fixed.

The natural action of $S_3$ on $\{1,2,3\}$ is faithful because any non-identity permutation does not leave all elements of $\{1,2,3\}$ fixed. Therefore, the action of $S_3$ on $\{1,2,3,4\}$ given by permuting the first $3$ elements is also faithful.

A group action is faithful if the homomorphism $G \to S_X$ is injective. This is the case for these actions, because different permutations in $S_3$ act "differently" on the elements of $\{1,2,3\}$ and $\{1,2,3,4\}$.