# Fully invariant direct factor

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: fully invariant subgroup and direct factor

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed a **fully invariant direct factor** if it satisfies the following equivalent conditions:

- It is both a fully invariant subgroup and a direct factor.
- It is both a homomorph-containing subgroup and a direct factor.
- It is both an isomorph-containing subgroup and a direct factor.
- It is both a quotient-subisomorph-containing subgroup and a direct factor.
- It is both a normal subgroup having no nontrivial homomorphism to its quotient group and a direct factor.

### Equivalence of definitions

`Further information: Equivalence of definitions of fully invariant direct factor`

## Examples

### Extreme examples

- Every group is a fully invariant direct factor in itself.
- The trivial subgroup is a fully invariant direct factor in every 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:

### Subgroups dissatisfying the property

Template:Subgroups dissatisfying property conjunction sorted by importance rank

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Sylow direct factor | Sylow subgroup that is also a direct factor | |||

Hall direct factor | Hall subgroup (order and index are relatively prime to each other) that is also a direct factor | |||

characteristic direct factor of abelian group | the whole group is an abelian group | Characteristic direct factor of nilpotent group|FULL LIST, MORE INFO | ||

characteristic direct factor of nilpotent group | the whole group is a nilpotent group | |FULL LIST, MORE INFO |