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

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, $N_{G}(P)\leq N_{G}(W(P))$ 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				|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

• Group in which the ZJ-functor controls fusion

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 $p$subgroup controls fusion in that subgroup, then $W$ controls fusion in the whole group.