# Number of orbits for finite group action on finite vector space equals number of orbits on dual vector space

Suppose $G$ is a finite group, $F$ is a finite field, and $V$ is a finite-dimensional vector space over $F$. Suppose we have a linear representation of $G$ on $V$, i.e., a homomorphism $G \to GL(V)$. The contragredient representation is a linear representation of $G$ on $V^*$, i.e., a homomorphism $G \to GL(V^*)$.
The claim is that the number of orbits of $V$ under the original $G$-representation equals the number of orbits of $V^*$ under the contragredient representation.