I have searched for such a question and didn't find it. I recently had a presentation in which I introduced $p$-Sylow subgroups and proved Sylow's theorems. I will have another one soon, concerning applications of Sylow's theorem.


My question is:



> 
> Are there any spectacular applications
>  of Sylow's theorem in group theory and
>  other fields of mathematics (which are
>  of course related to groups)?
> 
> 
> 



Applications for p-Sylow subgroups theorem
If you are introducing Sylow subgroups and the Sylow theorems, then your audience likely does not have an extensive mathematical background (otherwise I imagine they would have seen the Sylow theorems at some point in their studies, at least in North America and Western Europe). When I taught the Sylow theorems in an undergraduate abstract algebra class, I applied them to show converses of two basic properties of cyclic groups:


(1) If a finite group has at most one subgroup of each size then the group is cyclic. [Edit: There is a proof of this in the comments below which bypasses the Sylow theorems.]


(2) If a finite group has the property that for each positive integer $n$ the equation $x^n = 1$ has at most $n$ solutions in the group, then the group is cyclic.


In both proofs, you use the existence of $p$-Sylow subgroups to reduce yourself to the case of finite $p$-groups, and that case is then settled by other techniques (not using the Sylow theorems). Proofs of the above, together with other applications of the Sylow theorems can be found in my notes at <https://kconrad.math.uconn.edu/blurbs/grouptheory/sylowmore.pdf>.
Of course these are not "spectacular" applications, but I think it's cute that you can use the existence of Sylow subgroups to show either of those features of finite cyclic groups really characterize cyclic groups among all finite groups.


In a more advanced direction, Sylow subgroups are used to prove theorems about the cohomology of general finite groups. See Chapter IX of Serre's Local Fields (e.g., Theorems 12 and 13). This application is perhaps too much for your audience.


The Schur-Zassenhaus theorem about finite groups (see <https://en.wikipedia.org/wiki/Schur%E2%80%93Zassenhaus_theorem> for the statement) is proved using the Sylow theorems -- and not just the existence part of the theorems -- along with other techniques. I wrote up the simpler aspects of the proof in <https://kconrad.math.uconn.edu/blurbs/grouptheory/schurzass.pdf>.


A basic result about finite group actions is the Frattini argument: if a finite group $G$ acts on a finite set $X$ and a subgroup $H$ of $G$ acts transitively on $X$ then for every $x$ in $X$ we have $G = HS\_x = S\_xH$, where $S\_x$ is the stabilizer of $x$ in $G$. As an example of this, fix a prime $p$ and a finite group $G$. Let $K$ be a *normal* subgroup of $G$ and use for $X$ the set of all $p$-Sylow subgroups of $K$ (not of $G$!), which is a set on which $G$ acts by conjugation since $K$ is normal in $G$. That $K$ acts transitively on $X$ is a special case of the conjugacy part of the Sylow theorems (for the group $K$). Then Frattini's argument tells us that for any $p$-Sylow subgroup $P$ of $K$, we have $G = KN\_G(P)$, where $N\_G(P)$ is the normalizer subgroup of $P$ in $G$, since $N\_G(P)$ is the stabilizer of the "point" $P$ in the conjugation action of $G$ on $X$. This special case of the Frattini argument (which I think was the original version of Frattini himself)
can be used to show the equivalence of several different characterizations of finite nilpotent groups. It might be hard to convince students new to the Sylow theorems that this special case of the Frattini argument is a "spectacular" thing, but you ought to find it in any text on finite groups.


Finally, I think it would be good to place some of the basic features of the Sylow theorems in a broader context. I have in mind the following: existence (for any $p$ there is a $p$-Sylow subgroup), extension (any $p$-subgroup lies in a $p$-Sylow subgroup), and conjugacy (any two $p$-Sylow subgroups are conjugate). These aspects of $p$-Sylow subgroups for a fixed prime $p$ occur in other classes of groups, such as the maximal tori in connected compact Lie groups or connected linear algebraic groups. In the article "A Lie approach to finite groups" (see
<https://link.springer.com/chapter/10.1007/BFb0100726>), Alperin sets out an analogy between Lie groups and finite groups. See the table on page 4. In particular, he notes that Borel subgroups of Lie groups are analogous to normalizers of Sylow subgroups of a finite group.


