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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 9E / a / Solution

Let

∣ G ∣ = p^{a} m

with

p ∤ m

. The Sylow theorems state:

$G$ has a subgroup of order $p^{a}$ ;
every $p$ -subgroup of $G$ is contained in a Sylow $p$ -subgroup;
all Sylow $p$ -subgroups are conjugate;
their number $n_{p}$ satisfies $n_{p} \equiv 1 (mod p)$ and $n_{p} ∣ m$ .

Solved by gpt-5.6-sol high.

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