Fusion Systems
A fusion system is a category, whose objects are all subgroups of a finite p-group P, and whose morphisms are injective homomorphisms. The main example of a fusion system is a Sylow p-subgroup P of a finite group G, with the morphisms being given by conjugation of subgroups of P by elements of G. There are a few extra axioms that such a category should satisfy to be called a fusion system: broadly speaking, the conjugation by elements of the p-group itself should be in any fusion system; the concept of isomorphism (a bijective morphism between subgroups) should be an equivalence relation on the objects; and you should always be able to restrict the codomain of any map to the image.
Fusion systems themselves are faily loose objects and there is relatively little one can say about all fusion systems. One is naturally led to the concept of saturated fusion systems, which are fusion systems that satisfy a couple of extra axioms. Although the definition of saturation is fixed, there are several equivalent descriptions of it, and different authors prefer different definitions. Group fusion systems are examples of saturated fusion systems, but notably not all saturated fusion systems are group fusion systems, and are called exotic.
There are two different definitions of a normal subsystem, given in [1] and [7]. In [5] I relate the two definitions, proving that they are almost the same. The two concepts are called 'weakly normal' and 'normal', since weakly normal subsystems are a slightly weaker notion than normal subsystems. The main theorem from [5] is that a weakly normal subsystem of a fusion system is (essentially) a normal subsystem with a p'-group of automorphisms on top. This has the corollary that a simple fusion system is the same thing, regardless of which definition of normal subsystem one takes.
Some other work on the theory of fusion systems is currently mostly regarding cleaning up the current corpus of results into a coherent theory, together with extending the known results to new domains. In [4], I unite all the known ZJ-theorems, and in [3], we simplify (among other things) the proof that the product of two strongly closed subgroups is strongly closed (first given in [2]), and provides the isomorphism theorems for fusion systems.
In 2016, Bob Oliver, Jason Semeraro and I classified [6] all reduced (which is a mild generalization of simple) fusion systems on p-groups that possess an elementary abelian subgroup of index p. This classification yielded many new exotic systems and unified several previously obtained classification theorems.
References
[1] Michael Aschbacher, Normal subsystems of fusion systems, Proc. Lond. Math. Soc. 97 (2008), 239–271. MathSciNet.
[2] Michael Aschbacher, The generalized Fitting subsystem of a fusion system, Mem. Amer. Math. Soc. 209 (2011), no. 986.
[3] David A. Craven, Control of Fusion and Solubility in Fusion Systems, J. Algebra 323 (2010), no. 9, 2429--2448. DOI, PDF
[4] David A. Craven, On ZJ-Theorems for Fusion Systems, preprint, March 2009. PDF
[5] David A. Craven, Normal Subsystems of Fusion Systems, J. Lond. Math. Soc. 84 (2011), 137--158. DOI, PDF
[6] David A. Craven, Bob Oliver and Jason Semeraro, Reduced fusion systems over p-groups with an abelian subgroup of index p: II, submitted. PDF
[7] Markus Linckelmann, Introduction to fusion systems, Group Representation Theory, EPFL Press, Lausanne, 2007, pp. 79–113. MathSciNet.