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

Let $a$ and $b$ generate $C_m$ and $C_n$. The order of $(a,b)$ in the direct product of groups isWhen $\gcd (m,n)=1$, this is $mn=|C_m\times C_n|$. Hence $(a,b)$ generates the whole product and

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 $r\ge 0$.