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
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| fully invariant direct factor | fully invariant subgroup and a direct factor | fully invariant implies characteristic | characteristic direct factor not implies fully invariant | characteristic direct factor|fully invariant direct factor}} |
| Hall direct factor | Hall subgroup that is a direct factor | equivalence of definitions of normal Hall subgroup shows that normal Hall subgroups are fully invariant. | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| characteristic central factor | characteristic subgroup as well as a central factor | direct factor implies central factor | follows from center is characteristic and the fact that the center is not always a direct factor | |FULL LIST, MORE INFO |
| IA-automorphism-invariant direct factor | IA-automorphism-invariant subgroup and a direct factor | follows from characteristic implies IA-automorphism-invariant subgroup | ||
| IA-automorphism-balanced subgroup | every IA-automorphism of the whole group restricts to an IA-automorphism of the subgroup. | (via IA-automorphism-invariant direct factor) | (via IA-automorphism-invariant direct factor) | |FULL LIST, MORE INFO |
| characteristic subgroup | invariant under all automorphisms | |FULL LIST, MORE INFO | ||
| direct factor | normal subgroup with a normal complement | |FULL LIST, MORE INFO |
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.