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

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Statement

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.

References

Journal references