Small doubling does not control tripling in a noncommutative group
= Small doubling does not control tripling in a noncommutative group
Let $H$ be a finite subgroup and choose $x$ with $H\cap xHx^{-1}=\{1\}$. For $A=H\cup\{x\}$, the set $A^2$ has size at most $3|H|+1$, while $A^3$ contains the double coset $HxH$ of size $|H|^2$. Thus bounded doubling alone gives no tripling bound in arbitrary groups.