# Normal subgroup whose automorphism group is abelian

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 automorphism group is abelian
View a complete list of such conjunctions

## Statement

A subgroup $H$ of a group $G$ is termed an normal subgroup whose automorphism group is abelian of $G$ if it satisfies both these conditions:

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
cyclic normal subgroup cyclic group and normal subgroup cyclic implies abelian automorphism group abelian automorphism group not implies abelian |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup contained in centralizer of derived subgroup contained in the centralizer of derived subgroup derived subgroup centralizes normal subgroup whose automorphism group is abelian Normal subgroup contained in centralizer of derived subgroup|FULL LIST, MORE INFO
normal subgroup contained in centralizer of derived subgroup contained in the centralizer of derived subgroup derived subgroup centralizes normal subgroup whose automorphism group is abelian |FULL LIST, MORE INFO
normal subgroup whose inner automorphism group is central in automorphism group normal subgroup that is also a group whose inner automorphism group is central in automorphism group |FULL LIST, MORE INFO
commutator-in-center subgroup its commutator with whole group is in its center (via normal subgroup contained in centralizer of derived subgroup) Normal subgroup contained in centralizer of derived subgroup, Normal subgroup whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO
commutator-in-centralizer subgroup its commutator with whole group centralizes it
hereditarily 2-subnormal subgroup every subgroup of it is 2-subnormal
class two normal subgroup normal subgroup and has nilpotency class at most two abelian automorphism group implies class two class two not implies abelian automorphism group Commutator-in-center subgroup, Normal subgroup contained in centralizer of derived subgroup, Normal subgroup whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO