Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 9E / b / i

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- Solution i

Solution

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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