Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 10G / b

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- Solution b

Solution

The Sylow theorems state that if

∣ G ∣ = p^{a} m

with

p ∤ m

, then:

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

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