Conjugacy functor that controls fusion

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This article defines a property that can be evaluated for a conjugacy functor on a finite group. |View all such properties

Definition

Suppose G is a finite group and p is a prime number. Suppose W is a conjugacy functor on the nontrivial p-subgroups of G. We say that W controls p-fusion in G if, for any p-Sylow subgroup P of G, P is a weak subset-conjugacy-determined subgroup inside N_G(W(P)).

(Note that P is contained in N_G(W(P)) because W(P) is normal in P by the conjugation-invariance property that conjugacy functors have to satisfy. In fact, N_G(P) \le N_G(W(P)) by the fact that conjugacy functor gives normalizer-relatively normal subgroup).

Relation with other properties

Stronger 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

Facts

  • Control of fusion is local: If W is a conjugacy functor such that the restriction of W to the normalizer of any non-identity psubgroup controls fusion in that subgroup, then W controls fusion in the whole group.