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# A Strategy.

The complete program of how to apply the procedures described in the previous section and when to supplement them with additional information about the group is given by the following strategy.

A rough guideline is given by Lemma 4.3: Start with a trivial pre-fusion map. Keep producing stronger pre-fusion maps by splitting and fusing until a fusion map is reached.

2. Refine the equivalence relation with RefineClassesFrame (see 4.5).
3. Apply CheckSylowFrame (see 4.8) and CheckNormalizerFrame (see 4.10) in appropriate places.
4. Apply ConcludeFrame (see 4.6).
5. Stop if all classes are singletons. Otherwise find a legitimation for an action of one of the two following kinds.
1. Split an -class into two (or more) parts and goto 2., or
2. Fuse two (or more) images f(m) and goto 4.

Suppose, for example, that it is known (from the list of conjugacy classes of elements of G) that G has exactly two conjugacy classes of elements of order 2. Then G has two conjugacy classes of subgroups of order two. Suppose further that we know a pre-fusion map such that there exactly two images f(m) of size two and these two form one -class. Then, if the equivalence is derived from by splitting that class in two, then is a stronger pre-fusion map than . Moreover, this splitting will, via RefineClassesFrame, have an effect on all classes of subgroups that contain subgroups of order 2, depending on the distribution of the two types of subgroups of order two within them.

Suppose that the intersections of two classes of maximal subgroups are known, that one can find two classes and where and are in fact identical as subgroups of G, i.e. where . Suppose further that we know a pre-fusion map such that . Then, if f' is derived from f by fusing the images of and , then is a stronger pre-fusion map than . Moreover, this fusion will, via ConcludeFrame, have an effect on all classes which contain subgroups of or , because they also must fuse in some way. For examples of intersections of maximal subgroups (and methods how to determine them) see e.g. [Komissartschik and Tsaranov 1986].

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