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


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*What are some fundamental results in group theory?


We happened to list out some:


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*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.


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 $n$.
