Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 1 / i

Table of contents
- Solution i

Solution

For every

x \in B C^{- 1}

, choose one representation

x = b_{x} c_{x}^{- 1}

F : A \times B C^{- 1} ⟶ A B^{- 1} \times A C^{- 1}, F (a, x) = (a b_{x}^{- 1}, a c_{x}^{- 1}) .

The image determines

(a b_{x}^{- 1})^{- 1} (a c_{x}^{- 1}) = b_{x} c_{x}^{- 1} = x,

and the chosen representative then determines

a = (a b_{x}^{- 1}) b_{x}

. Thus

F

is injective, and counting its domain and codomain proves the Noncommutative Ruzsa triangle inequality

∣ A ∣ ∣ B C^{- 1} ∣ \leq ∣ A B^{- 1} ∣ ∣ A C^{- 1} ∣.

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