Correspondence between Linear Group Actions and Linear Representations

Theorem
Let $(k,+,\cdot)$ be a field.

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

Let $(G, *)$ 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 $\rho : G \to GL(V)$ 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 (g, v) = \rho(g)(v) $.

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 $V$.

Coversely suppose that $\phi : G \times V \to V$ is a linear action of $G$ on $V$, and define $\tilde \phi : G \to GL(V)$ by


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

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.