 Solution

Identify the additive group $G$ with the finite field ${\mathbb{F}}_q$, where $q=p^{\gamma }$. The sets $A$ and $B$ each have size $k<p$, and the field has odd characteristic. The Snevily matching theorem for an elementary abelian group states precisely that for two $k$-element subsets of the additive group of such a field, there is a bijection $\pi :A\to B$ for which the sums $a+\pi (a)$ are pairwise distinct.

For context, its polynomial proof antisymmetrizes the Vandermonde polynomialover all orderings of $B$. The coefficient furnished by the Dyson constant-term identity is a nonzero multiple of $k!$; it cannot vanish because $k<p$. Therefore at least one ordering $(b_1,\dots ,b_k)$ makes the Vandermonde product nonzero, which says exactly that