# Image-closed characteristic subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is termed an **image-closed characteristic subgroup** if, under any surjective homomorphism, its image is a characteristic subgroup of the image.

### Definition with symbols

A subgroup of a group is termed **image-closed characteristic** in if, for any normal subgroup of , is a characteristic subgroup of .

## Formalisms

### In terms of the image condition operator

This property is obtained by applying the image condition operator to the property: characteristic subgroup

View other properties obtained by applying the image condition operator

## Relation with other properties

### Stronger properties

### Weaker properties

### Related properties

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

### Image condition

YES:This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property

View other subgroup properties satisfying image condition

### Intermediate subgroup condition

NO:This subgroup property doesnotsatisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition