Characteristically complemented characteristic subgroup

From Groupprops
Jump to: navigation, search
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
View other subgroup property conjunctions | view all subgroup properties

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

Weaker 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: |
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 are groups such that is a characteristically complemented characteristic subgroup of and is a characteristically complemented characteristic subgroup of . Then is a characteristically complemented characteristic subgroup of . 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).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties