This chapter is concerned with various ways to prove that a fusion system, or a subsystem of a fusion system, is saturated. Finding out whether a given system is saturated is one of the most important, and one of the most difficult, aspects of the theory of fusion systems, and it will be useful to have different equivalent conditions, each tailored to a specific need.

The first section introduces the surjectivity property, a condition on the automorphisms of a given subgroup that, when combined with another condition on extensions of maps between subgroups, yields saturation. The power of this condition lies in its combination with the subsystem being invariant, which we will see later.

The second section introduces the notion of ℋ-saturation, which was considered by Broto, Castellana, Grodal, Levi, and Oliver in [BCGLO05]. This reduces the number of subgroups for which the saturation axioms need to be checked, basically to the class of centric subgroups. In fact, we can combine this result with the surjectivity property of the previous section to get another equivalent condition to saturation.

The third section moves in a different direction, describing the theory of invariant maps, as introduced by Aschbacher in [Asc08a]. Broadly speaking, they associate to each subgroup U of a strongly ℱ-closed subgroup Q a normal subgroup A(U) of Autℱ(U), in a consistent way. The subsystem generated by the subgroups A(U) will be an ℱ-invariant subsystem ε, although in general Autε(U) ≠ A(U) for some subgroups U of Q.