Group in which the ZJ-functor controls fusion for a prime

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Definition

Suppose $G$ is a finite group and $p$ is a prime number. We say that $G$ is a group in which the ZJ-functor controls fusion for $p$ if the ZJ-functor, viewed as a conjugacy functor on $G$, controls fusion in $G$ with respect to $p$. In other words, given a $p$-Sylow subgroup $P$ of $G$ and two subsets $A,B$ of $P$ that are conjugate in $G$, the subsets $A,B$ are conjugate in $N_G(Z(J(P))$.

Relation with other properties

Stronger properties

• Group of Glauberman type for a prime
• Group in which every p-local subgroup is of Glauberman type