 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/19h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write an element as

(x, y, z)

(x, y, z) (x^{'}, y^{'}, z^{'}) = (x + x^{'}, y + y^{'}, z + z^{'} + x y^{'}) .

The centre consists of

(0, 0, z)

. Each central element is its own conjugacy class. If

(x, y) \neq = (0, 0)

x, y

fixed and changes

z

by every element of

F_{p}

, so its class is

{(x, y, z) : z \in F_{p}} .

Thus there are $p$ central singleton classes and $p^{2} - 1$ noncentral classes of size $p$ .

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