Fully normalized potentially fully invariant subgroup

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 $H$ of a group $G$ is termed a fully normalized potentially fully invariant subgroup of $G$ if it satisfies both these conditions:

1. $H$ is a fully normalized subgroup of $G$, i.e., every automorphism of $H$ arises as the restriction to $H$ of an inner automorphism of $G$.
2. $H$ is a potentially fully invariant subgroup of $G$, i.e., there exists a group $K$ containing $G$ such that $H$ is a fully invariant subgroup of $K$.

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