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In particular, I'm just curious whether there's a version of the Sylow theorems (which are very combinatorially-flavored) which allows horizontal and/or vertical categorification? Or at least can be stated in more category-theoretic terms?

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I was wondering this just last week. I have prelims to start worrying about, and categorical statements always stick best. –  alekzander Nov 16 '09 at 0:49
    
alexzander, that's my justification too. I never did like abstract algebra 'til I learned to start thinking of it in categorical terms. –  Harrison Brown Nov 16 '09 at 3:06
    
Although in my case it's not prelims so much as undergraduate-level abstract algebra test, so I'm probably overthinking it. Still curious about the question on its own merits, though. –  Harrison Brown Nov 16 '09 at 3:15

5 Answers 5

Sylow subgroups are an example of a type of object satisfying a sort of universal property. Exploring other objects with similar properties gave birth to the modern theory of finite soluble groups in the 1960s.

If X is a class of finite groups and G is a finite group, then P is an X-covering subgroup of G if P is in X, and whenever P ≤ H ≤ G, N ⊴ H, and H/N in X, then PN=H. In other words, P covers every X-factor of G. If X is the class of finite p-groups, then X-covering subgroups of G and Sylow p-subgroups of G are the same thing. Indeed, if P is an X-covering group, and if H is a Sylow p-subgroup containing P and N=1, then we must have H=P. If P is a Sylow p-subgroup and H contains P with N ⊴ H and [H:N] a power of p, then [H:NP] is a divisor of [H:N] and [H:P], so must be 1. Notice how the "containment" part of the Sylow theorems is replaced with a "covering" condition that behaves better with the normal structure of the group.

If G is a finite solvable group and X is the class of nilpotent groups, then there is a sort of "Sylow nilpotent subgroup", the X-covering groups or Carter subgroups. They were studied by R.W. Carter who described them as self-normalizing nilpotent subgroups. Like Sylow p-subgroups, there is exactly one conjugacy class of Carter subgroups, and they have some reasonable arithmetic properties. People tried to determine which classes X of groups are such that X-covering groups exist and are unique up to conjugacy. Roughly speaking, this was the dawn of the modern theory of finite soluble groups, with Gaschütz's (et al.) classification of such X as "saturated formations".

This shifts focus away from the subgroup P to the class X. If X is sufficiently nice, then there will be a nicely embedded X-subgroup for any finite group.

Sylow p-subgroups also satisfy a dual condition, they are also X-injectors for the class X of finite p-groups. If X is a class of finite groups, and G is a finite group, then P is an X-injector of G if for every subnormal subgroup N of G, P∩N is a maximal X-subgroup of N. The dual definition of covering group (for X a saturated formation) is that P is an X-covering group iff PN/N is a maximal X-subgroup of G/N for every N ⊴ G. If X is the class of finite nilpotent groups, then X-injectors are called Fischer subgroups and again form a single, well-behaved conjugacy class of subgroups. A Fischer subgroup of a finite soluble group is a nilpotent subgroup that contains every nilpotent subgroup that it normalizes. This is similar to the idea that a Sylow p-subgroup contains every p-group that it normalizes. X such that X-injectors form a unique conjugacy class are called Fitting classes, due to their similarity to Fitting's lemma on subnormal nilpotent subgroups.

A very approachable introduction to these ideas is B.F. Wehrfritz's tiny textbook for a Second Course on Group Theory. Some of these ideas are described in Robinson's textbook for a Course in the Theory of Groups, but I believe it spends very little time on general formations. The standard textbook source for formations, especially in the soluble universe, is K. Doerk and T. Hawkes's book Finite Soluble Groups. Doerk&Hawkes explains several of Gaschütz's arithmetically defined Xs, which you might find a good contrast.

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This may be splitting hairs, but I think the Sylow theorems are more arithmetic than combinatorial. Depending on your standards of good categorification, I think this makes it difficult to encode precise congruence behavior such as the index of the normalizer being congruent to 1 mod p, and so on.

There is a homotopy-theoretic way to look at the Sylow theorems involving fusion systems. These are categories that encode the conjugation data between the p-Sylow subgroups of a finite group, and let you study the homotopy type of the p-completed classifying space of a finite or compact group. Bob Oliver has some papers on this subject. I don't know if this level of sophistication will help you learn the basic theorems, though. When I was studying Sylow, I personally preferred exercises, like showing that there are no simple groups of order 56.

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I learned the Sylow theorems precisely like that. Homework problem in my abstract algebra class "Show that there are no nonabelian simple groups of order less than 60." –  Charles Siegel Nov 16 '09 at 6:16
    
Yeah, okay, good point; it's probably too much to hope for there to be something easy enough to apply in day-to-day situations. But there's no obvious reason (to me, anyway) that there shouldn't be an extension even to finite groupoids. –  Harrison Brown Nov 16 '09 at 16:09
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J.L. Alperin's "Sylow Intersections and Fusion" (MR0215913; DOI: 10.1016/0021-8693(67)90005-1) is a nice group theoretic introduction to some of the key ideas behind fusion. It is a more precise version of Sylow's conjugacy theorem. Linckelmann's introduction (MR2336638) is one of the easiest ways to read the category theoretic version of fusion. Alperin's Local Representation Theory textbook introduces the Brauer subpairs theory nicely; that is the other main source of fusion systems. –  Jack Schmidt Feb 5 '10 at 4:21

Philosophically I think of the Sylow theorems as a statement about "localization of groups" at a prime. For abelian groups one can make this precise, since abelian groups, as Z-modules, can be realized as sheaves over Spec Z and the stalk over p is precisely the Sylow p-subgroup (although for finitely generated abelian groups these are trivial to describe). The hard part would be trying to carry this picture over to general groups.

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You might find the theory of fusion categories interesting (this is, unfortunately for the terminology, disjoint from Scott's mention of "fusion systems"). A fusion category is a type of tensor category which generalizes both Rep(G) and Vec_G, the latter being the category of vector spaces graded by a group G (so it's Groethendieck ring is the group algebra kG). These are tensor categories which are semi-simple and have finitely many isomorphism classes of simple objects. Each simple object has a notion of dimension, called Frobenius Perron dimension, and the categories themselves have a dimension associated to them, such that in the two cases Rep(G) and Vec_G, the dimension of the category is |G|.

While I don't know of Sylow theorems for fusion categories per se, there are many familiar theorems, saying e.g. that every fusion category of order p^k is nilpotent, and classifying groups of small orders p, pq, pqr, p^k, etc. There is also a version of Burnside's theorem stating that every category of dimension p^aq^b is solvable. It's quite possible that there are some analogs of Sylow's theorems in this direction.

I'd recommend "On Fusion Categories" http://arxiv.org/abs/math/0203060, by Etingof Nikshych and Ostrik, which introduces many of these ideas. More extensively, there are course notes from a class Etingof taught on this, which can be found on his webpage, http://math.mit.edu/~etingof. Papers by various subsets of (these three, union Gelaki) introduce the concepts and claims I mentioned above.

As Scott mentioned, this kind of abstract nonsense won't help much with prelims questions, most likely, but might make for a fun distraction =].

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One of the simplest consequences of Sylow's first theorem is the Cauchy theorem, saying that a group of order divisible by a prime $p$ contains an element of order $p$. I'd like to point out that there is an analog of Cauchy's theorem for semisimple Hopf algebras, arXiv:math/0311199, which has been generalized to quasi-Hopf algebras (i.e., fusion categories with integer dimensions) in arXiv:math/0601012.

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