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Next: Sylow's Theorem. Up: Approximating the fusion map. Previous: Fusion maps and pre-fusion

Subgroups of subgroups.

Let tex2html_wrap_inline3903 be a pre-fusion map on tex2html_wrap_inline3905 .

Define, for subgroups U and A of G, the set tex2html_wrap_inline3913 as the set of all subgroups B of U which lie in the tex2html_wrap_inline3919 -class of A. Denote by tex2html_wrap_inline3923 its size. This number is determined as follows. If U is a subgroup of the maximal subgroup tex2html_wrap_inline3927 of G, then (by Proposition 1.4)

displaymath3889

where the sums range over all representatives B of conjugacy classes of tex2html_wrap_inline3933 with tex2html_wrap_inline3935 .

  Prop640

proof645

This property can be used to detect elements of tex2html_wrap_inline3963 which are not conjugate in G.

  Cor648

proof654

A procedure which systematically applies Corollary 4.5 to tex2html_wrap_inline4019 in order to produce the strongest possible pre-fusion map from tex2html_wrap_inline4021 will be called RefineClassesFrame.

Comparing the sets S(U, A) for conjugate subgroups U leads to even more detailed insights.

Let tex2html_wrap_inline4027 and let tex2html_wrap_inline4029 . Let

displaymath3891

be the image of S(U, A) under f. Note that by the definition of S(U, A) all elements tex2html_wrap_inline4037 lie in the same tex2html_wrap_inline4039 -class. To each tex2html_wrap_inline4041 we associate a number tex2html_wrap_inline4043 defined by

displaymath3892

where the sum runs over the conjugacy classes of subgroups [B] of tex2html_wrap_inline4047 with tex2html_wrap_inline4049 . Conjugation in G partitions the set F(U, A) such that the sum of the tex2html_wrap_inline4055 corresponding to one part gives the number of G-conjugate subgroups of U of a certain type.

  Cor665

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, tex2html_wrap_inline4085 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 tex2html_wrap_inline4095 .

A procedure which systematically searches tex2html_wrap_inline4097 for places where Corollary 4.6 can be applied in order to produce the strongest possible pre-fusion map from tex2html_wrap_inline4099 will be called ConcludeFrame.


next up previous
Next: Sylow's Theorem. Up: Approximating the fusion map. Previous: Fusion maps and pre-fusion

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