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:
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.