= 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 polynomial
$$
\prod_{i<j}\bigl((a_i+x_i)-(a_j+x_j)\bigr)
$$
over 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 $(b_1,\ldots,b_k)$ makes the Vandermonde product nonzero, which says exactly that
$$
\boxed{a_1+b_1,\ldots,a_k+b_k\text{ are pairwise distinct}.}
$$
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