Let be a pre-fusion map on .
Define, for subgroups U and A of G, the set as the set of all subgroups B of U which lie in the -class of A. Denote by its size. This number is determined as follows. If U is a subgroup of the maximal subgroup of G, then (by Proposition 1.4)
where the sums range over all representatives B of conjugacy classes of with .
This property can be used to detect elements of which are not conjugate in G.
A procedure which systematically applies Corollary 4.5 to in order to produce the strongest possible pre-fusion map from will be called RefineClassesFrame.
Comparing the sets S(U, A) for conjugate subgroups U leads to even more detailed insights.
Let and let . Let
be the image of S(U, A) under f. Note that by the definition of S(U, A) all elements lie in the same -class. To each we associate a number defined by
where the sum runs over the conjugacy classes of subgroups [B] of with . Conjugation in G partitions the set F(U, A) such that the sum of the corresponding to one part gives the number of G-conjugate subgroups of U of a certain type.
This last result appears to be not as explicit as one might wish. In many cases, however, it is possible to explicitly determine the partitions and thus a new map f'.
If, for example, contains only one element, then, regardless of the size of F(U', A), the partition will be trivial and f' can be defined as f'(s') = s for each s' such that .
A procedure which systematically searches for places where Corollary 4.6 can be applied in order to produce the strongest possible pre-fusion map from will be called ConcludeFrame.