Different Applications of Sylow Theorems

I know exactly one other direct application of the Sylow theorems outside of group theory, which is to proving the fundamental theorem of algebra.

Suppose $K$ is a Galois extension of $\mathbb{R}$. We'll aim to show that either $K = \mathbb{R}$ or $K = \mathbb{C}$. (In particular, $\mathbb{C}$ itself must therefore be algebraically closed.) Let $G$ be its Galois group and let $H$ be the Sylow $2$-subgroup of $G$.

By Galois theory, $K^H$ is an odd extension of $\mathbb{R}$. But $\mathbb{R}$ has no nontrivial odd extensions: any such extension has primitive element something with an odd degree minimal polynomial over $\mathbb{R}$, but any such polynomial has a root by the intermediate value theorem. Hence $K^H = \mathbb{R}$, or equivalently $H = G$, so $G$ has order a power of $2$.

But now $K$ is an iterated quadratic extension of $\mathbb{R}$, and it's easy to explicitly show using the quadratic formula that the only nontrivial quadratic extension of $\mathbb{R}$ is $\mathbb{C}$, which itself has no nontrivial quadratic extensions.

The fundamental theorem of algebra is probably the best example, but how about the following proof that a finite subgroup of the multiplicative group of a field is cyclic:

Let $G \subset F^\times$ be finite. Let $H_p$ be a $p$-Sylow subgroup. Then if $|H_p|=p^k$, we claim that $H_p$ is simply the set of all roots in $F$ of the polynomial $x^{p^k}-1$: every element of $H_p$ is a root by LaGrange's theorem, and there can be no more roots since the degree of the polynomial is $p^k$. Now using this, it's easy to see that $H_p$ is cyclic: it's generated by any $y \in H_p$ such that $y^{p^{k-1}}\neq 1$ (such $y$ exist because there are only $p^{k-1}$ roots of $x^{p^{k-1}}-1$). But now since $G$ is abelian, $G$ is the direct product of its Sylow subgroups (this is where a Sylow theorem is used - although you could also use abelian group theory...), which are all cyclic of relatively prime order.

The Sylow Theorems often play a crucial role in finding all groups of a certain order. For example, all groups of order $pq$, or all groups of order $p^n$, where $p$ and $q$ are primes can be found in this manner. You may find more information in this book by J.S. Milne, chapter 5.