# Group in which every normal subgroup is characteristic

From Groupprops

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

### Symbol-free definition

A **group in which every normal subgroup is characteristic** is a group such that every normal subgroup of the group is characteristic in the group.

### Definition with symbols

## Formalisms

### In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (normal subgroup) satisfies the second property (characteristic subgroup), and vice versa.

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The property can be viewed as the following subgroup property collapse: normal subgroup = characteristic subgroup

### 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 (transitively normal subgroup), and vice versa.

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The property can be viewed as the following collapse: satisfies the property if and only if whenever is embedded as a normal subgroup of some group , then is a transitively normal subgroup of .

### In terms of the automorphism property collapse operator

This group property can be defined in terms of the collapse of two automorphism properties. In other words, a group satisfies this group property if and only if every automorphism of it satisfying the first property (automorphism) satisfies the second property (normal automorphism), and vice versa.

View other group properties obtained in this way

## Relation with other properties

### Stronger properties

- Group in which every normal subgroup is fully characteristic
- Simple group
- Cyclic group
- Complete group
- Group in which every automorphism is inner

### Weaker properties

- Group in which every normal subgroup is potentially characteristic: Here, every normal subgroup is potentially characteristic