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

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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]

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Definition

Suppose $K\leq G$ is a subgroup. We say that $K$ is a subgroup in which every subgroup characteristic in the whole group is characteristic if, for any subgroup $H\leq K$ such that $H$ is a characteristic subgroup of $G$, $H$ is also a characteristic subgroup of $K$.

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 transitive
ABOUT 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