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.

someapplications to theorems which don't look any more interesting than Sylow. The question, in the form "why should I care for Sylow's theorems if I am not a group theorist", is something I thought about asking some time ago. I am definitely NOT voting to close this. $\endgroup$ – darij grinberg Apr 4 '11 at 22:3610more comments