TITLE: Is there a geometric idea behind Sylow's theorems?
QUESTION [57 upvotes]: I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and geometric motivation for groups, except hastily mentioned dihedral groups), so when I (in self-study) discovered interplay between geometry and group theory, I was delighted. Many concepts and ideas suddenly made more sense to me.
I see Sylow's theorems as an useful technical black box, which can help you characterize groups when you only know numerical data about them. I've solved enough problems using those theorems, and now I'm interested if there is a way to make them ''click''.
I'm interested is there a geometrical idea behind Sylow's theorems (at least one of them), or at least a nice intuitive explanation of why that result should hold.
How do you think about them?

REPLY [4 votes]: I highly recommend going to a particular text: 
Visual Group Theory by Nathan Carter. 
He brings group concepts to life by using (colored!) Cayley diagrams. Chapter 9 deals specifically with Sylow theory. He gives many geometric arguments.
On amazon:
https://www.amazon.com/gp/aw/d/088385757X/ref=mp_s_a_1_sc_1?ie=UTF8&qid=1475779425&sr=8-1-spell&pi=AC_SX236_SY340_FMwebp_QL65&keywords=vosual+group+thwory