# Strongly closed conjugacy functor

From Groupprops

This article defines a property that can be evaluated for a conjugacy functor on a finite group. |View all such properties

## Contents

## Definition

Suppose is a finite group, is a prime number, and is a conjugacy functor for the prime . We say that is a **strongly closed conjugacy functor** if it satisfies the following equivalent conditions:

- There exists a -Sylow subgroup of such that is a strongly closed subgroup of relative to .
- For every -Sylow subgroup of , is a strongly closed subgroup of relative to .

## 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 | is a normal subgroup of . | |FULL LIST, MORE INFO |

### Weaker properties

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

weakly closed conjugacy functor | is weakly closed in relative to . | |FULL LIST, MORE INFO |