# Normal subgroup whose inner automorphism group is central in automorphism group

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## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): group whose inner automorphism group is central in automorphism group
View a complete list of such conjunctions

## Definition

A subgroup $H$ of a group $G$ is termed a normal subgroup whose inner automorphism group is central in automorphism group if $H$ is a normal subgroup of $G$ and $H$ is a group whose inner automorphism group is central in automorphism group.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
cyclic normal subgroup Normal subgroup whose automorphism group is abelian|FULL LIST, MORE INFO
abelian normal subgroup Normal subgroup whose automorphism group is abelian|FULL LIST, MORE INFO
normal subgroup whose automorphism group is abelian normal subgroup that is also a group whose automorphism group is abelian |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
commutator-in-center subgroup
class two normal subgroup Commutator-in-center subgroup|FULL LIST, MORE INFO
hereditarily 2-subnormal subgroup Commutator-in-center subgroup|FULL LIST, MORE INFO