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

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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 of a group is termed a **normal subgroup whose inner automorphism group is central in automorphism group** if is a normal subgroup of and 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 |