# Strongly 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, $p$ is a prime number, and $W$ is a conjugacy functor for the prime $p$. We say that $W$ is a strongly closed conjugacy functor if it satisfies the following equivalent conditions:

• There exists a $p$-Sylow subgroup $P$ of $G$ such that $W(P)$ is a strongly closed subgroup of $P$ relative to $G$.
• For every $p$-Sylow subgroup $P$ of $G$, $W(P)$ is a strongly closed subgroup of $P$ relative to $G$.

## 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 $W(P)$ is a normal subgroup of $G$. |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
weakly closed conjugacy functor $W(P)$ is weakly closed in $P$ relative to $G$. |FULL LIST, MORE INFO