# Group whose inner automorphism group is central in automorphism group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A **group whose inner automorphism group is central in automorphism group** is a group whose inner automorphism group is a central subgroup of the automorphism group. In other words, every inner automorphism of the group is a central automorphism.

## Formalisms

### In terms of the supergroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties in the following sense. Whenever the given group is embedded as a subgroup satisfying the first subgroup property (normal subgroup), in some bigger group, it also satisfies the second subgroup property (commutator-in-center subgroup), and vice versa.

View other group properties obtained in this way

A group is a group whose inner automorphism group is central in automorphism group iff, for any group containing as a normal subgroup, is a commutator-in-center subgroup of . `Further information: normal subgroup whose inner automorphism group is central in automorphism group is commutator-in-center`

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Abelian group | inner automorphism group is trivial | |FULL LIST, MORE INFO | ||

Group whose automorphism group is abelian | automorphism group is abelian | |FULL LIST, MORE INFO | ||

Cyclic group | generated by one element | (via abelian, also via abelian automorphism group) | 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 |
---|---|---|---|---|

Group of nilpotency class two | inner automorphism group is abelian | |FULL LIST, MORE INFO |