Normal fully normalized subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties:

normal subgroupProperty "Conjunction involving" (as page type) with input value "</br>normal subgroup" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.Property "Defining ingredient" (as page type) with input value "</br>normal subgroup" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process. and

fully normalized subgroupProperty "Conjunction involving" (as page type) with input value "</br>fully normalized subgroup" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.Property "Defining ingredient" (as page type) with input value "</br>fully normalized subgroup" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
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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 $\sigma \in \mathrm{A}\mathrm{u}\mathrm{t}(H)$ can be realized as conjugation by $g$ for some $g\in G$.

Equivalently, $H$ is normal in $G$, and the natural map $G\to \mathrm{A}\mathrm{u}\mathrm{t}(H)$ is surjective.

Relation with other properties

Conjunction with other properties

• NSCFN-subgroup is the conjunction with the property of being a self-centralizing subgroup.

Weaker properties

• Normal subgroup
• Fully normalized subgroup

Facts

• Every group is normal fully normalized in its holomorph
• Normal upper-hook fully normalized implies characteristic
• Characteristically simple and normal fully normalized implies minimal normal

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