# 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

## 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.