Complemented central factor
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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: permutably complemented subgroup and central factor
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- It is a permutably complemented subgroup as well as a central factor of the whole group.
- It is a complemented normal subgroup as well as a central factor of the whole group.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|direct factor||factor in an internal direct product||direct factor implies complemented central factor||complemented central factor not implies direct factor|||FULL LIST, MORE INFO|
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).
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Both the whole group and the trivial subgroup are complemented central factors.
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
If is a complemented central factor of a group , then is also a complemented central factor in any intermediate subgroup . This follows because both the property of being a permutably complemented subgroup and the property of being a central factor satisfy the intermediate subgroup condition. For full proof, refer: Permutably complemented satisfies intermediate subgroup condition, Central factor satisfies intermediate subgroup condition