Let and be as above. Moreover, let . In order to approximate the map we will work with idempotent maps on . An approximation of j will be called a pre-fusion map. Moreover, any map on that describes which elements of map into the same conjugacy class of subgroups of G will be called a fusion map. If we can determine one such fusion map f we can identify and and thus have found j = f.
For any pre-fusion map on we denote the set of all -classes on by and denote by the canonical map from to . Thus, for any , its -class is . The relation between the pair and the sets and is illustrated by the following diagram.
Due to (iii), we may regard as an equivalence relation on the set of all subgroups of G, with the property that whenever U and V are conjugate subgroups of G. Moreover, we may regard f as a function from the disjoint union of the sets of subgroups of the groups to via for any subgroup .
Note that, since is surjective and , also g must be surjective.
If, for example, is the identity map on and is the global equivalence on defined by for all then is a pre-fusion map since and, vacuously, implies .
The following lemma provides the termination condition of the approximation process.
The approximation process is guided by a partial order on the set of pre-fusion maps on .
Let and be pre-fusion maps on . We call stronger than (and write ) if there are maps and such that and , i.e. if the following diagram is commutative.
Let and be conjugate subgroups of G. Then . This allows us to define for . Thus, if we define an equivalence on by if , then is a pre-fusion map. (Because then, and implies .) Moreover, is stronger that , since, with for all and the identity map on we have .
Every group contains one trivial subgroup. There are r elements in corresponding to the trivial subgroups of the r maximal subgroups of G. Define
then is a pre-fusion map and is stronger than .