# Fully normalized potentially fully invariant subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: fully normalized subgroup and potentially fully invariant subgroup

View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup of a group is termed a **fully normalized potentially fully invariant subgroup** of if it satisfies both these conditions:

- is a fully normalized subgroup of , i.e., every automorphism of arises as the restriction to of an inner automorphism of .
- is a potentially fully invariant subgroup of , i.e., there exists a group containing such that is a fully invariant subgroup of .

## Relation with other properties

### Weaker properties

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

centralizer-annihilating endomorphism-invariant subgroup | fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant | |||

normal fully normalized subgroup | follows from potentially fully invariant implies normal |