# Complemented central factor

From Groupprops

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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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## Definition

A subgroup of a group is termed a **complemented central factor** or **split central factor** if it satisfies the following equivalent conditions:

- 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

### Stronger 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 |

### Weaker properties

## Metaproperties

### 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

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 conditionABOUT 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`