Symmetric Kneser's Theorem with Trios and $3$-Transform

We give a new equivalent restatement and a new proof in terms of trios to the classical Kneser’s theorem. In the finite case, our restatement takes the following, particularly symmetric shape: if $A$, $B$, and $C$ are subsets of a finite abelian group $G$ such that $A+B+C\mathrm{\ne }G$, then, denoting by $H$ the period of the sumset $A+B+C$, we have $$\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }+\mathrm{\mid }C\mathrm{\mid }\le \mathrm{\mid }G\mathrm{\mid }+\mathrm{\mid }H\mathrm{\mid }\mathrm{.}$$ The proof is based on an extension of the familiar Dyson transform onto set systems containing three (or more) sets.