= Free translation action on nontrivial subsets of a prime cyclic group
{title2=$|\{A+t:t\in\mathbb Z_p\}|=p\quad(0<|A|<p)$}
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 $p\mid\binom pk$ for $0<k<p$.
// Target: analysis.bigb
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