After using orbits and stabilizers to count how a group action moves elements of a set, we now extract a homomorphism from the action itself. Every element $g\in G$ sends each point $a\in S$ to another point $g\cdot a$, so it behaves like a permutation of $S$. This observation leads to permutation representations and extended Cayley theorem. A group action therefore converts an abstract group into a subgroup, or at least an image, inside a permutation group. In this lesson we shall prove the action-induced homomorphism, apply it to coset actions, and use the extended Cayley theorem to find normal subgroups.

Let $G$ be a group and let $S$ be a $G$-set. For each $g\in G$, define a function $\tau_g:S\to S$ by $$\begin{equation*} \tau_g(a)=g\cdot a. \end{equation*}$$ The assignment $$\begin{equation*} g\mapsto \tau_g \end{equation*}$$ is called the \textbf{permutation representation} associated with the action when it is viewed as a homomorphism from $G$ into the group of all permutations of $S$.