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 these groups, there are theorems saying 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 =].

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 =].