# Difference between revisions of "Weakly closed conjugacy functor"

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, $p$ a prime number, and $W$ a conjugacy functor on $G$ with respect to $p$. We say that $W$ is weakly closed in $G$ with respect to $p$ if the following equivalent conditions are satisfied:

1. Either of these equivalent:
• There exists a $p$-Sylow subgroup $P$ of $G$ such that $W(P)$ is a weakly closed subgroup of $P$ relative to $G$.
• For every $p$-Sylow subgroup $P$ of $G$, $W(P)$ is a weakly closed subgroup of $P$ relative to $G$.
2. Either of these equivalent:
• There exists a $p$-Sylow subgroup $P$ such that, for every $p$-Sylow subgroup $Q$ containing $W(P)$, $W(P) = W(Q)$.
• For every $p$-Sylow subgroup $P$, and for every $p$-Sylow subgroup $Q$ containing the center $W(P)$, $W(P) = W(Q)$.
3. Either of these equivalent:
• There exists a $p$-Sylow subgroup $P$ of $G$ such that for any $p$-Sylow subgroup $Q$ of $G$ containing $W(P)$, $W(P)$ is a normal subgroup of $Q$.
• For every $p$-Sylow subgroup $P$ of $G$, it is true that for any $p$-Sylow subgroup $Q$ of $G$ containing $W(P)$, $W(P)$ is a normal subgroup of $Q$.

For instance, a p-normal group is a group in which the conjugacy functor that arises by taking the center is weakly closed.