# Weakly 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

## Definition

Suppose is a finite group, a prime number, and a conjugacy functor on with respect to . We say that is **weakly closed** in with respect to if the following equivalent conditions are satisfied:

- Either of these equivalent:
- There exists a -Sylow subgroup of such that is a weakly closed subgroup of relative to .
- For every -Sylow subgroup of , is a weakly closed subgroup of relative to .

- Either of these equivalent:
- There exists a -Sylow subgroup such that, for every -Sylow subgroup containing , .
- For every -Sylow subgroup , and for every -Sylow subgroup containing , .

- Either of these equivalent:
- There exists a -Sylow subgroup of such that for any -Sylow subgroup of containing , is a normal subgroup of .
- For every -Sylow subgroup of , it is true that for any -Sylow subgroup of containing , is a normal subgroup of .

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

### Equivalence of definitions

`Further information: equivalence of definitions of weakly closed conjugacy functor`