# Characteristic p-functor that gives a characteristic subgroup

From Groupprops

This article defines a property that can be evaluated for a characteristic p-functor in the context of a finite group.|View other such properties

## Definition

Suppose is a prime number and is a finite group such that is a conjugacy functor for for the prime arising from a characteristic p-functor. We say that is a **characteristic p-functor that gives a characteristic 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 characteristic subgroup of .
- For every -Sylow subgroup of , is a characteristic subgroup 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`