I had been looking lately at Sylow subgroups of some specific groups and it got me to wondering about *why* Sylow subgroups exist. I'm very familiar with the proof of the theorems (something that everyone learns at the beginning of their abstract algebra course) -- incidentally my favorite proof is the one by Wielandt -- but the statement of the three Sylow theorems still seems somewhat miraculous. What got Sylow to imagine that they were true (especially the first -- the existence of a sylow subgroup)? Even the simpler case of Cauchy's theorem about the existence of an element of order $p$ in a finite subgroup whose order is a multiple of $p$ although easy to prove (with the right trick) also seems a bit amazing. I believe that sometimes the hardest part of a proving a theorem is believing that it might be true. So what can buttress the belief for the existence of Sylow subgroups?



Sylow Subgroups
Victor, you should check out Sylow's paper. It's in Math. Annalen 5 (1872), 584--594. I am looking at it as I write this. He states Cauchy's theorem in the first sentence and then says "This important theorem is contained in another more general theorem: if the order is divisible by a prime power then the group contains a subgroup of that size." (In particular, notice Sylow's literal first theorem is more general than the traditional formulation.) Thus he was perhaps in part inspired by knowledge of Cauchy's theorem. 


Sylow also includes in his paper a theorem of Mathieu on transitive groups acting on sets of prime-power order (see p. 590), which is given a new proof by the work in this paper. Theorems like Mathieu's may have led him to investigate subgroups of prime-power order in a general finite group (of substitutions). 


