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

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