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

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Solution

Composition and inversion in the Affine group of the complex line are

f_{a, b} \circ f_{a^{'}, b^{'}} = f_{a a^{'}, a b^{'} + b}, f_{a, b}^{- 1} = f_{a^{- 1}, - a^{- 1} b} .

The map

ρ : G ⟶ C^{\times}, ρ (f_{a, b}) = a

is therefore a surjective group homomorphism with kernel

N = {f_{1, b} : b \in C} ≅ (C, +) .

Its target is Abelian, so

[G, G] \subseteq N

. On the other hand, the stated computation gives

[f_{a, 1}, f_{1, b}] = f_{1, b (1 - a^{- 1})} .

Fixing any

a \neq = 1

and varying

b

produces every translation. Thus

N \subseteq [G, G]

, and hence

[G, G] = N ≅ (C, +), G / [G, G] ≅ C^{\times}, π (f_{a, b}) = a .

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