Permutation Representation of Group Action
Given a group action of $G$ on some set $A$, there is a natural way of representing $G$ in terms of permutations. Indeed, define the map $f_g:A \to A$ by $a \mapsto ga$. For injectivity, the associativity of group actions implies that
\[ga = gb \implies (g^{-1} g) a = (g^{-1} g) b \implies a = b.\]Surjectivity follows by noting that $g^{-1}a \mapsto a$ under $\sigma$, again, by associativity. It follows that $f_g$ is a permutation and, so, there is a natural map $f:G \to S_A$ given by $g \mapsto f_g$. This map is also a group homomorphism since
\[\begin{align*} f(gh) = f_{gh} = f_g f_h = f(g)f(h). \end{align*}\]We call $f$ the associated permutation representation. Thus, there are some natural definitions we can associate to the action.
Definition. The kernel of the $G$-action on $A$ is the kernel of the associated permutation representation.
This immediately motivates attention to certain kinds of permutation representations. In particular, if $f$ is not injective, then we lose some information about $G$ because two distinct elements of $g$ could map to the same permutation. Actions that have a trivial kernal are called faithful.
Example ($S_n$-action on $[n]$). Consider the canonical action of $S_n$ on $[n]$. This action is faithful because the associated permutation representation $S_n \to S_n$ is just the identity map.
Example ($D_{2n}$). Recall that $D_{2n}$ is given by the group presentation
\[D_{2n} = \langle r, s \mid r^n = s^2 = rsrs = 1\rangle.\]If we think of $[n]$ as a clockwise labeling of a regular $n$-gon in the plane, then there is a natural permutation representation of $D_{2n}$ obtained by setting
\[r \mapsto (1 \; 2 \; \cdots \; n) \quad\text{ and }\quad s = (2 \; n)(3 , n-1)\cdots.\]This representation is intuitively faithful because corresponds with our natural geometric intuition of $D_{2n}$.
Remark. Notice that the permutation representation gives rise to a natural group representation of dimension $|G|$ whose character corresponds to the number of fixed points.