Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 19I / b / i

Table of contents
- Solution i

Solution

Put

A = (0 ε ε 0), B = (ω 0 0 ω^{2}) .

Then

A^{2} = ε^{2} I

, so

A^{2 n} = I

because

ε^{2 n} = 1

, and

B^{3} = I

. Moreover,

A

exchanges the two eigenspaces of

B

A^{- 1} B A = B^{- 1} .

The matrices therefore satisfy every defining relation of

G_{6 n}

, and the universal property of a group presentation gives the required complex representation.

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