Free translation action on nontrivial subsets of a prime cyclic group ($|\{A+t:t\in {\mathbb{Z}}_p\}|=p(0<|A|<p)$)

If a nonempty subset $A$ of ${\mathbb{Z}}_p$ is invariant under translation by a nonzero $t$, then repeated translation of any member runs through all $p$ residues, so $A={\mathbb{Z}}_p$. Thus all $p$ translates of a nontrivial subset are distinct. Partitioning the $k$-element subsets into translation classes of size $p$ proves $p\mid \left(\genfrac{}{}{0ex}{}{p}{k}\right)$ for $0<k<p$.

// Target: analysis.bigb