# Conjugacy functor that controls strong 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, further, that $W$ is a conjugacy functor on $G$. We say that $W$ controls strong fusion on $G$ if, for any $p$-Sylow subgroup $P$ of $G$, $P$ is a subset-conjugacy-determined subgroup inside $N_G(W(P))$. In other words, given two subsets $A$ and $B$ in $P$ that are conjugate by $g \in G$, there exists $h \in N_G(W(P))$ such that conjugation by $h$ has the same effect as conjugation by $g$ on every element of $A$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions