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
- 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:
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|transitive subgroup property||Yes||Follows by combining characteristicity is transitive and direct factor is transitive||Suppose are groups such that is a characteristic direct factor of and is a characteristic direct factor of . Then, is a characteristic direct factor of .|
|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 , both the whole group and the trivial subgroup are characteristic direct factors.|
Relation with other 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|