Characteristic direct factor

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: [[conjunction involving::characteristic subgroup]][[defining ingredient::characteristic subgroup| ]] and [[conjunction involving::direct factor]][[defining ingredient::direct factor| ]]
View other subgroup property conjunctions | view all subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed a characteristic direct factor if it is a characteristic subgroup as well as a direct factor.

Relation with other properties

Stronger properties

• Fully characteristic direct factor
• Hall direct factor

Weaker properties

• Characteristic central factor

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

Since both the property of being characteristic and the property of being a direct factor are transitive, so is the property of being a characteristic direct factor.

In fact, it is a t.i. subgroup property.

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

Again, since both the property of being characteristic and the property of being a direct factor are trim, so is the property of being a characteristic direct factor.