Let $G$ be a group acting on a set $X$. For all $ g\in G $, we consider the application \begin{align*} \varphi_{g}: &~ X \longrightarrow X \\ &~ x \longrightarrow g.x \end{align*} It is clear that $\varphi_{gh}=\varphi_g\circ\varphi_h$, $\forall~ g,h \in G,$ $\varphi_e=\text{Id}_X$ ( $e$ the neutral element of $G$ ), and $\varphi_g \circ \varphi_{g^{-1}}=\varphi_{g^{-1}} \circ \varphi_g $, so $\varphi_g$ is bijective, for all $g\in G$. i.e: $\varphi \in \mathcal S(X)$, $\forall~ g\in G$, and the application: \begin{align*}\Phi :&~ G \longrightarrow \mathcal S(X) \\ &~g\longrightarrow \varphi_g\end{align*} is a group homomorphism that we call a representation of $G$ in $\mathcal S(X).$
I didn't understand what this definition is for, I'm looking for the intuition or the idea behind it. Why do we call the application $\Phi$ by this name: "representation", it is only a homomorphism? if anyone has any ideas or comments that he can add, I will be very grateful.