# Normal fully normalized subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and fully normalized subgroup
View other subgroup property conjunctions | view all subgroup properties

## Definition

### Symbol-free definition

A subgroup of a group is termed normal fully normalized if it is normal in the whole group, and every automorphism of the subgroup extends to an inner automorphism of the whole group.

### Definition with symbols

A subgroup $H$ of a group $G$ is termed normal fully normalized if $H$ is normal in $G$, and every $\sigma \in \operatorname{Aut}(H)$ can be realized as conjugation by $g$ for some $g \in G$.

Equivalently, $H$ is normal in $G$, and the natural map $G \to \operatorname{Aut}(H)$ is surjective.

## Metaproperties

### Left realization

Every group can be realized as a normal fully normalized subgroup in some group, for instance, in its holomorph. Further information: Every group is normal fully normalized in its holomorph