Importance of Cayley's theorem

Cayley's Theorem was very important historically. Groups originally arose from considering groups of permutations (specifically, the action of some functions on the roots of some polynomials). Every group was really a collection of permutations of some set of specific objects.

Then Cayley introduced the notion of an "abstract" group; the idea that you simply had some things that you could "compose" in some way giving you an associative operation, with an "identity" and inverses. He points out that all the things that people had been considering up to that point were "groups" in this sense. But he also wanted to make the point that he was not introducing a new class of objects, but that every object that satisfied his definition would also be a "group" in the old sense. That is, all he was doing was to recast the old notion, rather than expanding the class (at least, as applied to finite groups; for infinite groups there was no consensus on what "permutation of an infinite set" meant [whether it meant what we now think of as "bijection", or whether it meant what we now call "a bijection with finite support", that is, that only moves finitely many objects]). What Cayley's Theorem says is "Every thing that satisfies this definition can be thought of as a "group" in the old sense of a collection of permutations that is closed under composition, inverses, and includes the identity."

The idea of the proof turns out to be useful in other contexts as well, as indicated by T. So I would classify Cayley's theorem as more historically important than important today.

Well, it gives (part of) a proof of the first of Sylow's theorems: it is quite easy to prove that if $G$ is a finite group that admits a $p$-Sylow subgroup $S$, and if $H$ is a subgroup of $G$, then $H$ also admits a $p$-Sylow subgroup ($H$ acts on $G/S$, whose cardinal is prime to $p$, so there must be an orbit of cardinal prime to $p$, and $\mathrm{Stab}(gS)=H \cap gSg^{-1}$, so you find that the $\mathrm{Stab}$ of any element of this orbit is a $p$-Sylow of $H$).

Now, using Cayley's theorem, you can embed any finite group $H$ in $G=\mathrm{GL}_n(\mathbb{Z}/p \mathbb{Z})$ (think permutation matrices), where $n= \mathrm{card} H$, and $G$ admits a $p$-Sylow: the subgroup of upper-triangular unipotent matrices (recall $\mathrm{card} G = (p^n-1)(p^n-p) \ldots (p^n-p^{n-1})=p^{n(n-1)/2} m$ with $m$ prime to $p$).

It shows that the axiomatization of transformation groups is correct. Any transformation group is an axiomatic group, and any axiomatic group -- a structure, or more precisely a "model", satisfying the group axioms -- is a group of transformations. Historically, concrete groups of transformation appeared before abstract groups defined by algebraic axioms, so there is a question whether the algebra captures all (or possibly, more than) the intended examples.

Also, Cayley's theorem with its initially strange but retroactively instinctive idea of considering a structure's internal "action on itself", promoting symmetries of an object to the status of an object in their own (collective) right, is an early example of the formal style typical of later algebra and it helps prepare the ground psychologically for working in an abstract or axiomatized mode that was new in the 19th century.