 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ia/paper-3/1d/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

The centre is

Z (G) = {z \in G : z g = g z for every g \in G} .

H \leq Z (G)

is normal, since

g h g^{- 1} = h

for all

g \in G

and

h \in H

Write

D_{2 n} = ⟨ r, s : r^{n} = s^{2} = 1, srs = r^{- 1} ⟩

. A central rotation must satisfy

r^{k} = r^{- k}

, and no reflection is central for

n \geq 3

. Hence

Z (D_{2 n}) = {{1}, {1, r^{n /2}}, n odd, n even .

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