Fusion systems are structures that encode the properties of conjugation between p-subgroups of a group, for any prime number. It is always possible to define the saturated fusion system realized by a finite group on one of its Sylow p-subgroups. However, not all saturated fusion systems can be realized in this way: when this is the case, we say that the fusion system is exotic. The understanding of the behaviour of exotic fusion systems (in particular at odd primes) is still an important open problem.
