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)?
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.
You can prove the Fundamental Theorem of Algebra using Sylow 2-subgroups. The sketch of the proof is as follows:
That said, students will most likely encounter Sylow p-groups before Galois Theory. But that's not really an argument against providing the proof of FTA as a 'spectacular' application - especially if students are familiar with some basic field theory. The Galois theory it uses is in any case very elementary. And of course the key step is to show that $\vert$Gal$(k/\mathbb{R})\vert = 2^m$ which cannot be done without Sylow.
If anything this application illustrates exactly what a course on group theory should illustrate which is that a lot of analytic-flavoured questions involve insights that are much more adequately generalized in the framework of algebraic structures. (Of course some elementary analytic facts are still required in the proof.)
An application that I like, but it is by no means spectacular, is the following. Any finite p-group $G$ is isomorphic to a group of upper triangular matrices with ones on the diagonal (unitriangular matrices) over $F_p$.
Pf.
A counting argument shows the unitriangular group is a $p$-Sylow subgroup of the general linear group over $F_p$.
The symmetric group embeds in the general linear group via permutation matrices.
$G$ is isomorphic to a subgroup of a symmetric group.
Applying Sylow's theorem that all p-subgroups can be conjugated into a given p-Sylow subgroup completes the proof.
Of course this can be proved in other ways, but my students always seem to buy this as a nice application of Sylow.
Question: is there a regular map in the genus 2 surface with three heptagons meeting at each vertex? Answer: no. Proof: if there were, it would (using Euler's formula) need to have 28 vertices, 12 faces, and 42 edges, and therefore a rotational symmetry group of order 84. Now consider the Sylow-7-subgroups of this group. There must be 1 modulo 7 of them, and the only possibility is 1 of them. So the rotation that rotates and fixes one of the heptagons must rotate and fix all the heptagons. This can't work, so no such regular map can exist.
I'd like to give an answer which is less technical than some of the answers given so far, and more general. One of the main questions in group theory, informally, is just to describe the possible finite groups. "Describe" could be mean classify, and of course there is by now the fantastic Classification of Finite Simple Groups. But let's say that that wasn't yet known --- after all its proof is incredibly complicated. In any case not every finite group is simple. What could a group with 15 elements look like, say? Or a group with 100 elements? If you try to answer questions like this cold, most of them are very hard. Of course, you have various creative methods to give examples of finite groups that do exist: symmetric groups, dihedral groups, matrix groups, etc. However, it is surprisingly difficult to get started with the question of what groups do NOT exist. You would like some structure theorems about a mysterious finite group that someone else may or may not find, and not theorems about explicit finite groups that you make.
One of the few ways to get started is with the Sylow theorems, together with result that every p-group is nilpotent. (The second result follows from the lemma that every p-group has a center.) Sylow's theorems say that if you have a group with 15 elements, then it has a subgroup with 5 elements, and that subgroup is normal. It also has a subgroup with 3 elements, and with a tad more work that I will skip, there is only one group with 15 elements, the cyclic group. Now, finding the groups with 15 elements is not all that exciting, but how else were you going to analyze the groups with 15 elements or any other number of elements? In the specific case of 15 elements, there is Cauchy's theorem that there exists an element of every prime divisor order; but Sylow's theorems are best understood as a much stronger version of Cauchy's theorem.
A semi-separate way to get started is by observing that every finite group has a composition series with simple composants. This reduces the problem of classifying finite groups to classifying finite simple groups, and the problem of composing them together. But then what? To begin to classify finite simple groups, you are back to Sylow's theorems. Soon enough you need other tools, but as Keith Conrad explains, most of these tools use Sylow's theorems directly, while the others use them indirectly.
The above discusses why Sylow's theorems are important in group theory. A different question is why the theorems are important in the rest of mathematics. Occasionally people in other areas have the same classification concerns as group theorists. For instance, one question in 3-manifold topology is to classify which finite groups are fundamental groups of closed 3-manifolds. This question is now settled thanks to Geometrization and Perelman. However, the related question of which finite groups can act freely on a homology 3-sphere is still open, and it uses the same general classification theory.
But most of the time, mathematicians outside of group theory care much more about specific finite groups that exist than finite groups that do not exist. Since the Sylow theorems are at heart non-existence results, it is awkward to sell them to an audience that cares much more about the converse. Like if you want to buy a car, it would be strange for the salesman to brag about a technical paper that certain cars will never be built. Nonetheless, a wide variety of finite groups have turned out to be useful elsewhere in mathematics: In coding theory and combinatorial designs, in Galois theory, in string theory in the case of the monster group, etc. Many of the known finite groups were first considered simply as examples of finite groups that exist. Almost every finite simple group, in particular, has been used outside of group theory. Finite group theorists would not be very competent at finding finite groups for others to use, if they did not also have non-existence results.
To resonate with KConrad's answer, the first thing that came to my mind is Serre's proof in Cassels-Fröhlich: Algebraic Number Theory (Chapter VI) that for an extension $L/K$ of local fields, the abelian part of $\mathrm{Gal}(L/K)$ is isomorphic to $K^\times/N_{L/K}L^\times$. Perhaps this theorem can be mentioned in your lecture, the statement being beautiful and simple. An important technical step in the proof is the "Ugly Lemma", in Section 1.5, about the sizes of certain group cohomology groups, and the proof starts by reducing the statement to $p$-groups using the Sylow theorems.
Well, what constitutes a ``spectacular" application is a rather subjective judgement. The Classification of Finite Simple Groups has had many applications to other areas of Mathematics, and Bob Guralnick is one person who has highlighted many such applications, if anyone is inclined to search for explicit examples.
I do not believe that any experienced group theorist would argue that the Classification of Finite Simple Groups could have been achieved (at least the way it was) without Sylow's Theorem. For a group theorist, Sylow's Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing, and to stop and evaluate its applications takes some thought.
This is an over-simplification, but until the 1950's, there were not too many ways to prove that a finite group was not simple. There were a few character-theoretic results, and, for groups of small order, Sylow's theorem ( and the fact that the number of Sylow $p$-subgroups of a finite group was congruent to 1 (mod $p$)) was useful for ``small" orders (sometimes in an arithmetical sense, rather than cardinality). The main tool for proving non-simplicity was the transfer homomorphism. Theory was developed, some of it going back at least as far as Frobenius, to find conditions under which a finite group $G$ had a factor group of order $p$ for some prime $p$. These depended on the structure and embedding of the Sylow $p$-subgroup $P$ of $G$ (and sometimes the subgroups of $P$).
From the mid-50's, more non-simplicity criteria emerged, which used the structure of Sylow subgroups in different ways. The Theorem of Brauer and Fowler, which proved that there are only finitely many simple groups which have a centralizer of an involution with a given structure, and the Theorem of Brauer and Suzuki, which implied that the Sylow $2$-subgroup of any finite simple group (of even order) contains a Klein 4-group caused a shift of emphasis towards the structure of centralizers of involutions and of Sylow $2$-subgroups in finite simple groups. The Odd Order Theorem of Feit and Thompson, and later the $N$-group paper of John Thompson, began to reveal the true power of what Jon Alperin later termed ``local analysis"- the structure and embedding of $p$-subgroups and their normalizers, to unlock the structure of finite groups, simple groups in particular. The general principle which seemed to emerge, and was later exploited further for the whole classification, was that in the presence of sufficiently large elementary Abelian subgroups, local analysis (sometimes with respect to several primes, the favoured prime usually being $2$) could pin down the stucture of putative simple groups sufficiently to identify them ( of course, the work of Chevalley, Steinberg, Tits and others in obtaining elegeant unified descriptions of the known (non-sporadic) simple groups, was essential too).
Fortunately, techiques of ordinary and modular character theory were well-suited to identifying finite simple groups whose Sylow $2$-subgroups did not contain elementary Abelian subgroups of order $8$. Other techniques were developed to deal with extreme cases where group-theoretic information was restricted: one was Glauberman's $Z*$-theorem, proved using modular character theory, which proved that if a finite group $G$ had no non-trivial normal subgroup of odd order, and $S$ was a Sylow $2$-subgroup of $G$, then an involution $t \in S$ was in $Z(G)$ if and only if it was not conjugate in $G$ to any other element of $S$.
The work of George Glauberman, of John Thompson, and of Michael Aschbacher (in particular- there were other major contributors) showed the power of local group-theoretic analysis, as techniques were further honed and developed. So, an answer is, that Sylow's Theorem is an absolute keystone of finite group theory, and therefore implicitly underpins many applications of finite group theory as a whole to other areas of Mathematics.
Even the cyclicity of the groups of order 15, or the existence of a normal Sylow 5-subgroup in any group of order 100, is not merely a toy example.
The fact that Sylow p-subgroups of a finite group are always conjugate is one way to prove that normal implies characteristic for a Sylow p-subgroup. (So if a group has simple subgroups of index 100 which generate it, and no normal subgroup of order 25, the group itself is simple. Hence, the Higman-Sims group is simple because the Mathieu group $M_{22}$ is simple. This is done in Wilson's "The Finite Simple Groups".) Two consequences of this are that if $P$ is a Sylow p-subgroup of a finite group $G$ and $K$ is a subgroup satisfying $N_{G}(P) \leq K \leq G$, then $[K:N_{G}(P)] \equiv [G:K] \equiv 1 \mod{p}$ and $K$ is self-normalizing in $G$. In particular, the maximal subgroups of $G$ containing $N_{G}(P)$ are constrained by these results.
The cyclicity of groups of order 15 is more than just a toy example, since the cyclicity of groups of order 299 = 13*23 (which is provable the same way) is used in Thompson's original proof of the simplicity of the Conway group $Co_{1}$. (This proof also gives an example of the use of the Frattini argument.)
If you want to prove the Burnside $p^{a}q^{b}$-Theorem, you need to exploit the existence of Sylow subgroups. This is one of the few commonalities of the character-theoretic and character-free proofs of the theorem. Via character theory, the basic group-theoretic result is that a finite group with a conjugacy class whose size is a power of a prime cannot be simple -- but you can only get a conjugacy class of size equal to a power of a prime in a group of order $p^{a}q^{b}$ by choosing a nontrivial central element of a Sylow subgroup (unless you made a bad choice and it's in the center of the whole group, in which case nonsimplicity of the group is immediate unless the group is cyclic of prime order).
Eschewing character theory, Sylow subgroups are indispensable, whether you use the Glauberman $ZJ$-theorem or any other local-analytic tools to do the heavy lifting in the proof. They are also essential even for much lighter lifting which happens in these proofs.
When using the transfer to prove a finite group satisfying certain conditions is not perfect, it's good to have a subgroup from which this fact is visible. It is good to have a subgroup $H$ such that one knows $\phi : G \to A$ is nontrivial because its restriction to $H$ is nontrivial. If p is a prime dividing $| \phi(G) |$, then any subgroup whose index is a nonmultiple of p will work. A Sylow p-subgroup of $G$ fits the bill perfectly, and often comes with a fair amount of information about its own structure, to boot.
It is possible to build off of the Burnside $p^{a}q^{b}$-Theorem to prove the that the existence of Sylow systems characterizes finite solvable groups. Sylow system normalizers are all conjugate in a finite solvable group, and these facts form the starting point of the theory of finite solvable groups (which is substantial in its own right, as one can read in "Finite Soluble Groups", by Doerk and Hawkes).
Here is a proof of Wilson's theorem, that $(p-1)!\equiv -1\,\text{mod}\,p$.
Sylow's 3rd theorem implies that the symmetric group $S_p$ (order $p!$) has $1\,\text{mod}\,p$ subgroups $\mathbb{Z}/p\mathbb{Z}$ cyclic, and since they all only intersect at the 0 element ($p$ is prime) there are a total of $(p-1)\cdot1\,\text{mod}\,p\equiv-1\,\text{mod}\,p$ elements of order $p$, which equivalently are the $(p-1)!$ amount of $p$-cycles.