TITLE: How is the kernel of a group action defined?
QUESTION [6 upvotes]: Question:

Show that the kernel of the group action of $G$ acting on set $A$ is equal to the kernel of the corresponding permutation representation of this action. 

I'm lost in this definition as I am only familiar with the definition of a kernel of a homomorphism as $\{g \in G~|~\varphi(g) = I_A\}$ (the elements of the domain in which the image is the identity of the target group)
How is this definition applicable to a map $f: G\times A \to A$ in which the target domain is a set?

REPLY [7 votes]: Let $G$ be a group acting on $A$. 
The kernel of the action is the set $K =\{g \in G; g \cdot a = a , \forall a \in A\}$. Now the corresponding permutation representation is a group homomorphism $\psi : G \to S_A$ given by  $\psi (g)(a) = g \cdot a $.

$K \subseteq \ker \psi$

Let $k \in K$, then for all $a \in A$ we have that $\psi (k) (a) = k \cdot a = a$ thus, $\psi (k) = id_A$ and then $k \in \ker \psi$. 

$\ker \psi \subseteq K$

Let $k \in \ker \psi$ be given. Then for all $a \in A$  we have $$ k \cdot a = \psi (k) (a) = id_A(a) = a $$
thus $k \in K$. 
Edit: The kernel of $\psi$ is given by $\ker \psi = \{g \in G ; \psi (g) =id_A\}$.