The first equivalence follows from Correspondence Between Group Actions and Permutation Representations. === 1 implies 2 === Let $\rho : G \to \operatorname{GL} \left({V}\right)$ be a linear representation of $G$ on $V$. Then for all $g \in G$, $v_1, v_2 \in V$: {{begin-eqn}} {{eqn | l = \phi \left({g, v_1 + v_2}\right)       | r = \rho \left({g}\right) \left({v_1 + v_2}\right)       | c = {{Defof|Group Action Associated to Permutation Representation}} }} {{eqn | r = \rho \left({g}\right) \left({v_1}\right) + \rho \left({g}\right) \left({v_2}\right)       | c = $\rho \left({g}\right)$ is linear }} {{eqn | r =\phi \left({g, v_1}\right) + \phi \left({g, v_2}\right)       | c = {{Defof|Group Action Associated to Permutation Representation}} }} {{end-eqn}} and for all $g \in G$, $v \in V$, $\lambda \in k$: {{begin-eqn}} {{eqn | l = \phi \left({g, \lambda \cdot v }\right)       | r = \rho \left({g}\right) \left({\lambda \cdot v}\right)       | c = {{Defof|Group Action Associated to Permutation Representation}} }} {{eqn | r = \lambda \cdot \rho\left({g}\right) \left({v}\right)       | c = $\rho \left({g}\right)$ is linear }} {{eqn | r = \lambda \cdot \phi \left({g, v}\right)       | c = {{Defof|Group Action Associated to Permutation Representation}} }} {{end-eqn}} Therefore $\phi$ is a linear group action of $G$ on $V$. {{qed|lemma}} === 2 implies 1 === Let $\phi: G \times V \to V$ be a linear action of $G$ on $V$. Then for all $g \in G$, $v_1, v_2 \in V$: {{begin-eqn}} {{eqn | l = \rho \left({g}\right) \left({v_1 + v_2}\right)       | r = \phi \left({g, v_1 + v_2}\right)       | c = {{Defof|Permutation Representation Associated to Group Action}} }} {{eqn | r = \phi \left({g, v_1}\right) + \phi \left({g, v_2}\right)       | c = {{Defof|Linear Group Action}} }} {{eqn | r =  \rho \left({g}\right) \left({v_1}\right) + \rho \left({g}\right) \left({v_2}\right)       | c = {{Defof|Permutation Representation Associated to Group Action}} }} {{end-eqn}} and for all $g \in G$, $v \in V$, $\lambda \in k$: {{begin-eqn}} {{eqn | l = \rho \left({g}\right) \left({\lambda \cdot v}\right)       | r = \phi \left({g, \lambda \cdot v }\right)       | c = {{Defof|Permutation Representation Associated to Group Action}} }} {{eqn | r = \lambda \cdot \phi \left({g, v}\right)       | c = {{Defof|Linear Group Action}} }} {{eqn | r = \lambda \cdot \rho\left({g}\right) \left({v}\right)       | c = {{Defof|Permutation Representation Associated to Group Action}} }} {{end-eqn}} Therefore $\rho$ is a linear representation of $G$ on $V$. {{qed|lemma}} {{qed}} {{refactor|the below can safely be merged into Group Action defines Permutation Representation}} Now let $g_1, g_2 \in G$. We have for all $v \in V$: {{begin-eqn}} {{eqn | l = \rho \left({g_1 * g_2}\right) \left({v}\right)       | r = \phi \left({g_1 * g_2, v}\right)        | c = }} {{eqn | r = \phi \left({g_1, \phi\left({g_2, v}\right) }\right)       | c = $\phi$ is an action }} {{eqn | r = \rho \left({g_2}\right) \left({\rho \left({g_1}\right) \left({v}\right) }\right)       | c =  }} {{eqn | r = \left({ \rho \left({g_2}\right) \circ \rho \left({g_1}\right) }\right) \left({v}\right)        | c = where $\circ$ is the composition of mappings }} {{end-eqn}} Thus $\rho$ satisfies the homomorphism property. {{refactor|the below is better merged into the theorem statement}} Therefore: :$\hat{} : ($linear representations$) \to ($linear actions$)$ :$\tilde{} : ($linear actions$) \to ($linear representations$)$ give a bijection. {{qed}}