Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 149 / 1 / d / Solution

Choose a covering set $X$ of size at most $K$ such that $A^2\subseteq XA$, as allowed by the definition of an approximate group. Discard every $x\in X$ for which $xA$ does not meet $A^2$. Each remaining $x$ belongs to $A^2{A}^{-1}=A^3$, so $X\subseteq ⟨A⟩$. Induction givesBecause $A$ is symmetric and contains the identity, $⟨A⟩={\bigcup }_{m\ge 1}{A}^m$. Hence

We use the bounded-exponent finitely generated nilpotent group order bound. In an $s$-step nilpotent group, a subgroup generated by $k$ elements is generated in collected form by the simple group commutators in those generators of weights at most $s$. There are at mostsuch commutators. Every one has order at most $r$, soTaking $H=⟨A⟩$ now gives