Past exam of the short options course of the University of Oxford / 2017 / 2 / a / ii

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- Solution ii

Solution

Write the group order as

∣ G ∣ = p^{a} m

, where

p

is a prime number and

p ∤ m

. A Sylow p-subgroup is a subgroup of group order

p^{a}

, the largest power of

p

dividing

∣ G ∣

.

The three Sylow theorems assert:

Existence: a Sylow p-subgroup exists, and every subgroup of prime power order a power of $p$ is contained in one.
Conjugacy: any two Sylow p-subgroups are conjugate in $G$ .
Counting: their number $n_{p}$ satisfies $n_{p} ∣ m$ and $n_{p} \equiv 1 (mod p)$ .

In particular, a unique Sylow p-subgroup is a normal subgroup, since all its conjugates must equal it.

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