Characteristic direct factor
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: characteristic subgroup and 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: For proof of the implication, refer fully characteristic implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer characteristic direct factor not implies fully characteristic.
- Hall direct factor
Weaker properties
Facts
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.