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Specifically: If $p$ is a prime divisor of the order of a finite group $G$, then there exists an element of order $p$ in $G$

So I'm looking for a little intuition behind this idea. I understand how to prove it, but I don't understand the idea behind it. I.e, if I hadn't already seen a proof, I wouldn't know why or if the statement of the theorem was correct. As stated, I'm looking for the core idea behind the proof.

Any insights on how you think of it would be appreciated! Thanks!

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Could you put up your understanding of what the theorem is and what proof you understand? Different things get different names is different places sometimes, especially if it's with someone as prolific as Cauchy. – Tom Oldfield Nov 16 '12 at 0:17
I'm sorry, I thought Cauchy did most of his work in analysis and that the "for Groups" would suffice. My apologies – user45793 Nov 16 '12 at 0:19
I'm sure it would, just want to make sure everyone is one the same page! – Tom Oldfield Nov 16 '12 at 0:20
@Tom: I didn't have any problem recognizing that the title corresponds to the theorem that the OP has stated. I'm pretty sure that his title is specific enough. – Haskell Curry Nov 16 '12 at 0:26
@Carl: The proof that you have seen - is it the one by McKay that uses a group action? – Haskell Curry Nov 16 '12 at 0:29
up vote 3 down vote accepted

What is the "intuition: behind the theorem "a function differentiable at some point is continuous at that point"? I'm not sure, but perhaps it'd be that the function is "smooth" enough at that point as to be continuous...or something like that.

What's the intuition behind Cauchy's Theorem? That a finite group having order a multiple of a prime $\,p\,$ has "to pay the price", i.e.: it must have at least one element of order that prime $\,p\,$...or something like this.

I can't say what the core behind Cauchy's idea was, but perhaps it stemmed from checking many examples and seeing there was an apparent common pattern to all of them.

So I'm not sure, but perhaps it was Cauchy's Theorem what gave Sylow some ideas or inspiration to make some research on this and eventually to come up with some of the most important and basic theorems in finite group theory.

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The idea behind it is that it is a partial converse to Lagrange's Theorem (that the order of any subgroup divides the order of the group). We seeks theorems of this kind because it means we can get information about the elements of a group just from knowing it's order, and then we can start to classify groups. i.e. we can then say that if a group has a certain size, it will be of a certain form. This gives an intuitive reason why we want theorems of this kind, but not of why it is true.

In fact, I don't think the theorem is intuitive(which I know is not what you want to hear!), and the proof that I've seen involving having the $\mathbb{Z_p}$ act on the set of p-tuples of the group is something that seems totally unnatural (although once understood it is very pretty, not to mention clever!) In fact, I don't think that there is a core idea behind the proof to be help understand why it's true. I think it is just an exercise in clever manipulation, not giving us much insight into the problem, in the same way that inductive proofs often give us answers but may not help us understand the question better.

I think that theorems like this one (and also the Sylow theorems) aren't obvious, but they are very, very useful. As such, when groups were first discovered, people were desperately trying to find theorems of this sort, so spent a long time thinking about them and discovered them in some manner or another, and so they were born for utility, not because they were intuitively true.

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