# Conjugacy functor that gives a normal subgroup

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 prime number and is a finite group such that is a conjugacy functor for for the prime . We say that is a **conjugacy functor that gives a normal subgroup** if it satisfies the following equivalent conditions:

- For every pair of -Sylow subgroups of , .
- For every pair of -Sylow subgroups of , is a normal subgroup of .
- Each of these:
- is a weakly closed conjugacy functor and there exists a -Sylow subgroup of such that where is the p-core of .
- is a weakly closed conjugacy functor and for every -Sylow subgroup of , where is the -core of .

- Each of these:
- There exists a -Sylow subgroup of such that is a normal subgroup of .
- For every -Sylow subgroup of , is a normal subgroup of .

### Equivalence of definitions

`Further information: equivalence of normality and characteristicity conditions for conjugacy functor`

## Relation with other properties

### Weaker properties

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

conjugacy functor that controls strong fusion | element-wise conjugacy in a -Sylow subgroup is determined in the normalizer of | |FULL LIST, MORE INFO | ||

conjugacy functor that controls fusion | subset-wise conjugacy in a -Sylow subgroup is determined in the normalizer of | Conjugacy functor that controls strong fusion, Conjugacy functor whose normalizer generates whole group with p'-core|FULL LIST, MORE INFO | ||

conjugacy functor whose normalizer generates whole group with p'-core | For a -Sylow subgroup, | |FULL LIST, MORE INFO | ||

strongly closed conjugacy functor | returns a subgroup that is a strongly closed subgroup in the Sylow subgroup relative to the whole group. | |FULL LIST, MORE INFO | ||

weakly closed conjugacy functor | returns a subgroup that is a weakly closed subgroup in the Sylow subgroup relative to the whole group. | Strongly closed conjugacy functor|FULL LIST, MORE INFO |