Characteristic direct factor
From Groupprops
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
Contents
Definition
A subgroup of a group is termed a characteristic direct factor if it is a characteristic subgroup as well as a direct factor.
Examples
Extreme examples
- Every group is a characteristic direct factor of itself.
- The trivial subgroup is a characteristic direct factor in any group.
Subgroups satisfying the property
Here are some examples of subgroups in basic/important groups satisfying the property:
Here are some examples of subgroups in relatively less basic/important groups satisfying the property:
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | Yes | Follows by combining characteristicity is transitive and direct factor is transitive | Suppose ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
trim subgroup property | Yes | Follows from the fact that both the property of being characteristic and the property of being a direct factor satisfy this condition. | In any group ![]() ![]() |
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. | Characteristically complemented characteristic subgroup, Fully invariant direct factor|FULL LIST, MORE INFO |