Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 3 / d

Solution

Realize the generators as affine reflections of

R^{2}

s (x, y) = (- x, y), u (x, y) = (x, - y), t (x, y) = (1 - y, 1 - x) .

The reflecting lines for

s

and

u

are perpendicular, while the line

x + y = 1

meets each at angle

π /4

. Hence

s^{2} = t^{2} = u^{2} = (s u)^{2} = (s t)^{4} = (t u)^{4} = 1,

g = s u t, qq u a d g (x, y) = (y - 1, x - 1), qq u a d g^{2} (x, y) = (x - 2, y - 2),

and

g

W

W (B_{4})

is the signed symmetric group on four letters and has order

2^{4} 4!

. An infinite group cannot embed in it, so

W

is not a subgroup of

W (B_{4})

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