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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.

  1. Start with tex2html_wrap_inline4315 (see above) as the initial pre-fusion map.
  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 tex2html_wrap_inline4317 -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 tex2html_wrap_inline4329 such that there exactly two images f(m) of size two and these two form one tex2html_wrap_inline4333 -class. Then, if the equivalence tex2html_wrap_inline4335 is derived from tex2html_wrap_inline4337 by splitting that class in two, then tex2html_wrap_inline4339 is a stronger pre-fusion map than tex2html_wrap_inline4341 . 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 tex2html_wrap_inline4345 and tex2html_wrap_inline4347 where tex2html_wrap_inline4349 and tex2html_wrap_inline4351 are in fact identical as subgroups of G, i.e. where tex2html_wrap_inline4355 . Suppose further that we know a pre-fusion map tex2html_wrap_inline4357 such that tex2html_wrap_inline4359 . Then, if f' is derived from f by fusing the images of tex2html_wrap_inline4365 and tex2html_wrap_inline4367 , then tex2html_wrap_inline4369 is a stronger pre-fusion map than tex2html_wrap_inline4371 . Moreover, this fusion will, via ConcludeFrame, have an effect on all classes which contain subgroups of tex2html_wrap_inline4373 or tex2html_wrap_inline4375 , 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].


next up previous
Next: The Table of Marks Up: The Subgroups of Previous: Normalizers.

Götz Pfeiffer Wed Oct 30 09:52:08 GMT 1996