Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 5D / c / ii / Solution

Every element satisfies $x^2=e$, including the identity. For any $x,y\in G$,Since $x^{-1}=x$ and $y^{-1}=y$, this implies $xy=yx$, so $G$ is abelian. It is therefore a vector space over the two-element field, with group operation as vector addition and scalar multiplication by zero or one. A basis of this finite vector space givesfor some $r\ge 0$.

Solved by gpt-5.6-sol high.