Application of Cayley’s theorem in Sylow’s theorem

Question

I’ve just started reading Sylow’s theorems. I have heard that Cayley’s theorems are applied in Sylow’s theorem. Can someone exactly point out where in the three Sylow’s theorem is Cayley’s theorem explicitly used?

Answer 1

Only one of the several possible proofs of Sylow I uses Cayley's theorem and it's not a very common one (at least not in my experience; I never ran into it as a student). According to Waterhouse it's essentially due to Frobenius (I had access to this pdf and then lost it so unfortunately I am going off of memory and might be misremembering). It starts with the following lemma:

Lemma: If $H$ is a subgroup of a finite group $G$ such that $G$ has a Sylow $p$-subgroup $P$, then $H$ also has a Sylow $p$-subgroup, which is given by its intersection with some conjugate of $P$.

(Here a Sylow $p$-subgroup is a $p$-subgroup whose index is coprime to $p$. This is not equivalent to "maximal $p$-subgroup" until you prove Sylow I + II.)

Proof. Consider the action of $H$ on the cosets $G/P$. By hypothesis, $|G/P|$ isn't divisible by $p$, hence at least one orbit of this action also has size not divisible by $p$. This orbit contains some coset $gP$, and its size is $\frac{|H|}{|\text{Stab}(gP)|}$ where the stabilizer of $gP$ consists of all $h \in H$ such that

$$hgP = gP \Leftrightarrow g^{-1} hgP = P \Leftrightarrow g^{-1} hg \in P$$

and hence $\text{Stab}(gP) = (g^{-1} H g) \cap P$. So the stabilizer is a subgroup of $P$ and hence also a $p$-group, and also its index in $H$ is coprime to $p$, so we conclude that it's a Sylow $p$-subgroup of $g^{-1} Hg$. Conjugating, we get that $H \cap (gPg^{-1})$ is a Sylow $p$-subgroup of $H$. $\Box$

Now by Cayley's theorem, every finite group $H$ embeds into some symmetric group $S_n$. So you can prove Sylow I by proving that the symmetric groups have Sylow $p$-subgroups, and these can be constructed explicitly (although they're somewhat annoying to describe; if you want to try doing this you can reduce to the case that $n$ is a power of $p$ where it's easier). You can also embed $S_n$ into $GL_n(\mathbb{F}_p)$ and exhibit a Sylow $p$-subgroup there, which is just the upper triangular matrices with $1$s on the diagonal.

This argument, slightly simplified, gives an independent proof of Cauchy's theorem, which is essentially Cauchy's original proof of Cauchy's theorem; it ran $9$ pages. Note also that setting $H$ to be a $p$-group in the lemma immediately gives Sylow II, in the strong form that not only are all Sylow $p$-subgroups conjugate but every $p$-subgroup is a subgroup of a Sylow $p$-subgroup, so "Sylow" and "maximal" coincide.