Importance of Cayley's theorem

Question

I along with one of my friends were just discussing some basic things in group theory, when this question came up:

• What are some fundamental results in group theory?

We happened to list out some:

• Fundamental Theorem of Group Homomorphism:

• Cayley's Theorem

• Sylow's Theorem

There may be many more, but as far as my little knowledge is concerned, i think these are very important. Then we started explaining why each one of the above results were more powerful. I could explain as to how the Fundamental Theorem of Group Homomorphism can be used to derive some good results, in group theory, and also i could show my friends the power of the Sylow Theorems just by considering groups of order $pq$ $\bigl($ For e.g the case were $p \nmid (q-1)$ $\bigr)$. But i could never illustrate him as to how powerful Cayley's theorem is.

Can anyone explain the significance of Cayley's theorem and why it plays a central role in group theory. I am also curious to know whether any important results proved in Group theory using Cayley's theorem.

Answer 1

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.

Answer 2

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 $p$).

Answer 3

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.

Answer 4

It is central to the theory of group representations. It allows to prove that any group $G$ of cardinal $n$ has a representation (a vector space $E$ such that $G$ is isomorphic to the linear group of $E$) of dimension $n$.

Answer 5

Here are interesting facts or applications about Cayley's theorem:

If $G$ is a group of order $2.(odd)$ then $G$ has normal subgroup of index $2$.

The proof of this exactly uses the homomorphism in Cayley's theorem and a simple observation in it.

If $G$ is a finite group, then $G$ can be embedded in a finite group $\hat{G}$ such that two elements of $G$ which have same order become conjugate in $\hat{G}$.

Again, here $\hat{G}$ is symmetric group on $|G|$ symbols, and embedding is Cayley map.

This interesting property of embedding of groups making elements of same order conjugate was a topic for infinite groups, and for which, a positive answer came through well-celebrated HNN-extension theorem.

Answer 6

Not sure this is as "central" as asked, but Cayley embedding ("$\theta$") allows some intuition about the notion of isomorphism ("$f$") between two groups ("$G$, "$H$"). In fact, the operation-preserving property of the bijection $f$ is the necessary and sufficient condition to make this diagram commuting: $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\las}[1]{\kern-1.5ex\xleftarrow{\ \ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} & & \\ G & \ras{\space\space\space f \space\space\space} & H \\ \da{\theta_G} & & \da{\theta_H} \\ \operatorname{Sym}(G) & \ras{\psi^{(f)}} & \operatorname{Sym}(H) \\ \end{array} $$ where $\psi^{(f)}\colon \operatorname{Sym}(G)\to \operatorname{Sym}(H)$ is the bijection defined by $\sigma\mapsto f\sigma f^{-1}$; namely:

$$\theta_H=\psi^{(f)}\theta_Gf^{-1}$$

In words: after duly renaming of the elements of $G$ ("$f^{-1}$"), the structure of $G$ ("$\theta_Gf^{-1}$") can be "transported" into $\operatorname{Sym}(H)$ ("$\psi^{(f)}\theta_Gf^{-1}$") precisely onto the structure of $H$ ("$\theta_H=\psi^{(f)}\theta_Gf^{-1}$"). I think this is what is usually meant -for the finite case- with "comparing the Cayley tables of two groups".