# Subgroup in which every subgroup characteristic in the whole group is characteristic

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

Suppose is a subgroup. We say that is a **subgroup in which every subgroup characteristic in the whole group is characteristic** if, for any subgroup such that is a characteristic subgroup of , is also a characteristic subgroup of .

## Relation with other properties

### Stronger properties

- Direct factor
- AEP-subgroup
- Normal subgroup in which every subgroup characteristic in the whole group is characteristic
- Characteristic subgroup in which every subgroup characteristic in the whole group is characteristic
- Subgroup whose intersection with a characteristic subgroup of the whole group is characteristic in it

## 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