# Fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant

From Groupprops

## Statement

Suppose is a subgroup of a group satisfying the following two conditions:

- is a Fully normalized subgroup (?) of , i.e., every automorphism of extends to an inner automorphism of .
- is a Potentially fully invariant subgroup (?) of , i.e., there exists a group containing in which is a fully invariant subgroup.

Then, is a Centralizer-annihilating endomorphism-invariant subgroup (?) of : For every endomorphism of whose kernel contains , .