Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 8D / a / i / Solution

Identify $A$ with $\{(a,1):a\in A\}$. Componentwise use of the stated multiplication shows that this set contains the identity and is closed under products and inverses, so it is a subgroup. It has the two cosetsand therefore has index two. Every index-two subgroup is normal, so

Solved by gpt-5.6-sol high.