Cauchy's Theorem for Groups

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.

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.