Correspondence between Linear Group Actions and Linear Representations

Theorem
Let $\left({k, +, \cdot}\right)$ be a field.

Let $V$ be a vector space over $k$ of finite dimension.

Let $\left({G, *}\right)$ be a finite group.

Then there is a one-to-one correspondence between linear actions of $G$ on $V$ and linear representations of $G$ on $V$.

Proof
First suppose that $\rho : G \to \operatorname{GL} \left({V}\right)$ is a linear representation of $G$ on $V$.

Define $\hat\rho : G \times V \to V$ by:


 * $\forall g \in G, v \in V: \hat \rho \left({g, v}\right) = \rho \left({g}\right) \left({v}\right)$

Then for all $g \in G$, $v_1, v_2 \in V$:

and for all $g \in G$, $v \in V$, $\lambda \in k$:

Therefore $\hat \rho$ is a linear action of $G$ on $V$.

Conversely suppose that $\phi: G \times V \to V$ is a linear action of $G$ on $V$.

Define $\tilde \phi : G \to \operatorname{GL} \left({V}\right)$ by:


 * $\forall g \in G, v\in V: \tilde\phi \left({g}\right) \left({v}\right) = \phi\left({g,v}\right)$

Now let $g_1, g_2 \in G$.

We have for all $v \in V$:

Thus $\tilde \phi$ satisfies the homomorphism property.

Therefore:


 * $\hat{} : ($linear representations$) \to ($linear actions$)$
 * $\tilde{} : ($linear actions$) \to ($linear representations$)$

give a bijection.