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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.

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In response to your list, I am a big fan of the fundamental theorem of finitely generated abelian groups... but maybe you consider that a corollary. –  BBischof Nov 12 '10 at 18:36
    
Also, I find it strange. You care about groups of order pq, but what about other groups of specific order? If you do some exercises in Sylow theory, you will quickly find ones that require you to look at the induced automorphisms on subgroups, and they exploit Cayley's theorem and simplicity results of symmetric groups to get contradictions. If you are interested in groups of order pq, go look at classification of groups and how they use Cayley's. –  BBischof Nov 12 '10 at 18:38
    
@BBischof: Please post whatever you have in mind as an terse answer, so that many naive people may appreciate the question as well as the answer. –  anonymous Nov 12 '10 at 19:13
    
@BBischof: I consider groups of order $pq$ because, i know that if $p \not\mid (q-1)$, then $G$ is cyclic –  anonymous Nov 12 '10 at 19:17
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What about Lagrange's theorem? From what I've seen, I would say that's where it all begins.. –  Mikko Korhonen Dec 6 '11 at 21:02

4 Answers 4

up vote 9 down vote accepted

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 $p$).

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Very nice! (and some characters to fulfill the silly requirement of 15 characters) –  Mariano Suárez-Alvarez Nov 12 '10 at 19:24

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.

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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.

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+1. Cayley's theorem may be obvious in retrospect, but there are natural ways to define groups (e.g. the fundamental group or homology groups) that don't act on any obvious object, so it was difficult to recognize these as groups before the axiomatic definition. –  Qiaochu Yuan Nov 12 '10 at 18:48
    
@Qiaochu: I have trouble in understanding your comment! –  anonymous Nov 12 '10 at 19:14
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@Chandru1: those groups from topology (and many others not from topology) are defined by "generators and relations", not as transformations of any object. They are constructed as axiomatic groups but not as transformation groups. Cayley's theorem shows that they also belong to the latter class. –  T.. Nov 12 '10 at 19:19
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@Chandru1: people such as Riemann had been studying what were essentially homology groups before they were actually recognized as groups. The reason they weren't is that if you only think of groups as transformation groups, it's hard to see what the homology groups are transforming. Once the abstract definition of a group became well-known it was possible to see that homology wasn't just Betti numbers, it carried a group structure, and this was central to the development of algebraic topology. –  Qiaochu Yuan Nov 12 '10 at 19:22
    
Yes, i understood somewhat! Have not read about fundamental groups. Guess thats the problem –  anonymous Nov 12 '10 at 19:31

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 $n$.

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This is a reason, but it is not to my mind the reason. –  Qiaochu Yuan Nov 12 '10 at 18:45
    
faithful representation –  Plop Nov 12 '10 at 19:05
    
you're both right –  marwalix Nov 13 '10 at 7:24

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