Exercise 1.7.6 (Faithful action)

Proof. As said in the text (p.43):

“If $G$ acts on a set $A$ and distinct elements of $G$ induce distinct permutations of $A$, the action is said to be faithful. A faithful action is therefore one in which the associated permutation representation is injective.”

Since the kernel $\ker <!--\; FUNCTION\; APPLICATION\; -->(\phi )$ of the permutation representation is the same as the kernel $K$ of the action by Exercise 5, $G$ acts faithfully on a set $A$ if and only $\ker <!--\; FUNCTION\; APPLICATION\; -->(\phi )=\{1_G\}$, that is if and only if the kernel of the action $K=\{g\in G\mid \forall <!--\; FUNCTION\; APPLICATION\; -->a\in A,g\cdot a=a\}=\{1_G\}.$ □