# Complemented central factor

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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:

1. It is a permutably complemented subgroup as well as a central factor of the whole group.
2. 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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
complemented normal subgroup normal subgroup with a permutable complement Complemented transitively normal subgroup|FULL LIST, MORE INFO
complemented transitively normal subgroup transitively normal subgroup with a permutable complement |FULL LIST, MORE INFO
permutably complemented subgroup has a permutable complement (via complemented normal) (via complemented normal) Complemented normal subgroup|FULL LIST, MORE INFO
lattice-complemented subgroup has a lattice complement (via permutably complemented) (via lattice-complemented) Complemented normal subgroup|FULL LIST, MORE INFO
central factor product with centralizer is whole group (by definition) central factor not implies complemented |FULL LIST, MORE INFO
normal subgroup invariant under inner automorphisms (via central factor, also via complemented normal) (via central factor, also via complemented normal) Complemented normal subgroup, Complemented transitively normal subgroup, SCAB-subgroup|FULL LIST, MORE INFO

## 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 condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H$ is a complemented central factor of a group $G$, then $H$ is also a complemented central factor in any intermediate subgroup $K$. 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