Definition of Representation in terms of Group Action 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 $V$?
 A: What seems to have been confusing the asker (or lacking from the source they used) is the distinction between a permutation representation and a (linear) representation of a group.
In the former case the group $G$ acts on a set $X$ that may not (or may?) have some additional structure. Often the set $X$ is finite, when the action gives rise to a homomorphism of groups $\phi:G\to Sym(X)\cong S_n, n=|X|$. This allows us to use a permutation $\phi(g)$ to do calculations involving the group element $g$. Tools from combinatorics then come to the fore, and can be used to study the group $G$.
In the latter case we study homomorphisms $\phi:G\to GL(V)$, $V$ a vector space over a field $K$. Then $G$ acts on the vector space $V$, but we additionally insist that $G$ respects the vector space operations, i.e. the action is via $K$-linear transformations. The study of (linear) representations brings in tools from linear algebra to the study of the properties of a group.
There are other kinds of interesting group actions. If we want to study the action of a group $G$ on a topological space/a manifold/an algebraic variety $X$, then we naturally want the action to consist of bijections observing the extra structure, i.e. continuous/differentiable/rational respectively.
The various kinds of representations of $G$ are related. Of course, all the representations are permutations of the elements of $X$, but that is not necessarily very interesting (e.g. nearby points are often permuted by the exact same permutations not shedding additional light to the group $G$). However, nearly all the actions listed above give rise to interesting linear representations. The permutations can be turned into monomial matrices by interpreting the elements of $X$ as (formal) basis vectors, the actions on a topological space/manifold/variety lead to linear actions on the (co)homology spaces.
Also, the study of group representations is always a 2-way street. We use properties of $X$ to study $G$, and conversely we use properties of $G$ to study $X$. For example, the result known as (not) Burnside's lemma is a nice tool in combinatorics. And in the 60s-70s the physicists used properties of groups of symmetries of nature to deduce/predict what kind of elementary particles must exist.
A: 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.
