 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-3/8d/b/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The action on the four diagonals identifies

G ≅ C_{2} \times S_{4},

where the

C_{2}

factor is generated by central inversion. An element of order three must have trivial

C_{2}

component and a three-cycle in

S_{4}

. Since all three-cycles in

S_{4}

are conjugate,

all elements of order three in G are conjugate .

Elements of order two are not all conjugate. For example,

- I

is central and therefore has a singleton conjugacy class, while a reflection is a different element of order two. Thus

elements of order two in G need not be conjugate .

 New to topics? Read the docs here!