The definition of a representation of a group $G$ over a vector space $V$ is a map $p: G \to GL(V)$. According to wikipedia, for finite groups an equivalent definition is an action of $G$ on $V$. I'm having trouble seeing how these two definitions are equivalent.
An action of $G$ on $V$ means a map $G \times V \to V$ satisfying $(gh)v=g(hv)$ and $ev=v$. I see how a map $G \times V \to V$ determines a function from $G$ to the set of functions on $V$ - just fix a $g$ and the function $f_g$ is defined by $f_g(v)=gv$. Further, I see how the axioms for a group action mean that this function from $G$ to the set of functions on $V$ is a map: the compatibility axiom guarenteed that $f_h \circ f_g=f_{gh}$. However, I am struggling to see why the function $f_g$ from $V$ to $V$ is necessarily a linear transformation. How do we get from the set of functions on $V$ to the set of linear transformations on $V$?
Instead of an equivalence between representations and linear actions, I have found that the one induces the other (and conversely, too). The following mappings hopefully clarify the idea.
Suppose $\rho \colon G \to \mathrm{GL}(V)$ is a representation. Define $\phi \colon G \times V \to V$ by the map $(g, v) \mapsto \rho(g)v$ and prove it is a linear action. Note that since $\rho$ is a representation, by definition $\rho(g) \in \mathrm{GL}(V)$ for all $g \in G$. Using this linearity you can prove that $\phi$ is a linear action.
Conversely, suppose $\phi \colon G\times V \to V$ is a linear action and define $\rho \colon G \to \mathrm{GL}(V)$ by the map $g \mapsto \phi(g, \cdot)$, where $\cdot$ a placeholder for an element of $V$. To show that $\rho(g)$ is linear for all $g \in G$, use the fact that $\phi$ is a linear action: \begin{equation} \rho(g)(\lambda v + w) = \phi(g, \lambda v + w) = \lambda \phi(g,v) + \phi(g, w) = \lambda \rho(g)v + \rho(g)w. \end{equation} Conclude that $\rho(g)$ is indeed linear. So a representation induces a linear action, and a linear action induces a representation.