Conjugacy functor that controls fusion
This article defines a property that can be evaluated for a conjugacy functor on a finite group. |View all such properties
Suppose is a finite group and is a prime number. Suppose is a conjugacy functor on the nontrivial -subgroups of . We say that controls -fusion in if, for any -Sylow subgroup of , is a weak subset-conjugacy-determined subgroup inside .
(Note that is contained in because is normal in by the conjugation-invariance property that conjugacy functors have to satisfy. In fact, by the fact that conjugacy functor gives normalizer-relatively normal subgroup).
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|Conjugacy functor that gives a normal subgroup||Conjugacy functor that controls strong fusion, Conjugacy functor whose normalizer generates whole group with p'-core|FULL LIST, MORE INFO|
|Conjugacy functor that controls strong fusion|||FULL LIST, MORE INFO|
|Conjugacy functor whose normalizer generates whole group with p'-core||Conjugacy functor whose normalizer generates whole group with p'-core controls fusion|||FULL LIST, MORE INFO|
Related group properties
- Control of fusion is local: If is a conjugacy functor such that the restriction of to the normalizer of any non-identity subgroup controls fusion in that subgroup, then controls fusion in the whole group.