Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 19I / b / i / Solution

The subgroup $⟨a⟩$ is the unique Sylow-seven subgroup and is normal. No element outside it centralizes $a$, so $|C_G(a)|=7$. Similarly $C_G(b)=⟨b⟩$ has order three. Thus nonidentity powers of $a$ form two classes of size three, while the elements of orders three form two classes of size seven. Together with the identity, there are five conjugacy classes.

Solved by gpt-5.6-sol high.