Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 1 / ii / Solution

Choose $a_0\in A$. Right multiplication by $a_0$ injects $A^2$ into $A^3$, soApply part i with anchor $A^{-1}$, $B=A$, and $C=A^2$. Since inversion preserves cardinality,Therefore $|A{A}^{-1}{A}^{-1}|\le K^2|A|$.

Apply part i again, now with anchor $A$, $B=A^2$, and $C=A^{-1}{A}^{-1}$. This givesThus $|A^4|\le K^3|A|$. Since $K\ge 1$, this proves the requested fourfold product bound from small tripling $|A^4|\le K^4|A|$.