Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 149 / 3 / a / Solution

Composition and inversion in the Affine group of the complex line areThe mapis therefore a surjective group homomorphism with kernelIts target is Abelian, so $[G,G]\subseteq N$. On the other hand, the stated computation givesFixing any $a\ne 1$ and varying $b$ produces every translation. Thus $N\subseteq [G,G]$, and hence