Characteristically complemented characteristic subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: characteristically complemented subgroup and characteristic subgroup
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Definition

A subgroup of a group is termed a characteristically complemented characteristic subgroup if it is a characteristic subgroup and it has a permutable complement that is also a characteristic subgroup.

Relation with other properties

Stronger properties

• Hall direct factor
• Sylow direct factor

Weaker properties

• Left-transitively complemented normal subgroup
• Characteristic direct factor
• Characteristically complemented subgroup
• Complemented characteristic subgroup
• Complemented normal subgroup
• Retract
• Normal subgroup

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

Suppose $H\leq K\leq G$ are groups such that $H$ is a characteristically complemented characteristic subgroup of $K$ and $K$ is a characteristically complemented characteristic subgroup of $G$. Then $H$ is a characteristically complemented characteristic subgroup of $G$. For full proof, refer: Characteristically complemented characteristic is transitive

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).
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