# 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

## Contents

## Definition

A subgroup of a group is termed a **characteristic direct factor** if it is a characteristic subgroup as well as a direct factor.

## Examples

### Extreme examples

- 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:

## Metaproperties

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

### Stronger 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 |