 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-2/9e/b/i/solution

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

By transitivity and the orbit-stabilizer theorem,

∣ G ∣ = 7∣ G_{x} ∣ = 7 \cdot 24 = 168 = 2^{3} \cdot 3 \cdot 7.

Inside

G_{x} ≅ S_{4}

, there are three Sylow

2

-subgroups, each of order eight, and four Sylow

3

-subgroups, each of order three. These are also Sylow subgroups of

G

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