# Direct factor

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## Definition

QUICK PHRASES: factor in internal direct product, normal with normal complement, has centralizing complement

### Definition in tabular form

A **direct factor** of a group is defined in the following equivalent ways:

No. | Shorthand | A subgroup of a group is a direct factor if ... | A subgroup of a group is a direct factor of if ... |
---|---|---|---|

1 | factor in internal direct product | its internal direct product with some subgroup is the whole group | there is a subgroup of such that is the internal direct product of and |

2 | normal with normal complement | it is a normal subgroup with a normal complement, i.e., it is both a normal subgroup and a retract | is normal and there is a normal subgroup of such that the product and is trivial. |

3 | has centralizing complement | there is a subgroup centralizing it, intersecting it trivially, and whose product with it is the whole group | there is a subgroup of such that (where is the centralizer in of ), is trivial, and . |

4 | factor in internal direct product of multiple subgroups | it is one of the subgroups occurring in an internal direct product decomposition of the whole group into (possibly more than two) subgroups. Note that we also allow infinitely many subgroups, in which case, the internal direct product would correspond to the restricted external direct product. | there is a collection of subgroups with equal to one of the s, such that is the internal direct product of the s. |

### Equivalence of definitions

The equivalence of definitions follows largely from the equivalence of internal and external direct product.

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Direct factor, all facts related to Direct factor) |Survey articles about this | Survey articles about definitions built on this

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This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and retract

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## Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this propertyVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

### Extreme examples

Every group is the internal direct product of itself and the trivial subgroup. Thus:

- The trivial subgroup is a direct factor of the whole group.
- Every group is a direct factor of itself.

### High occurrence examples

- In a finite nilpotent group, all the Sylow subgroups are direct factors. In particular, a finite nilpotent group is the direct product of its Sylow subgroups.
`Further information: equivalence of definitions of finite nilpotent group` - In a vector space, any vector subspace is a direct factor, because the complementary subspace can be taken as the complement for an internal direct product.

### Relationship with external direct product and restricted external direct product

- If a group arises as the external direct product of finitely or infinitely many groups , then for any subset , the subset of arising as those elements where all coordinates outside of are trivial is a direct factor of . The complementary factor can be taken as the subgroup of where all coordinates in are trivial.
- A similar observation holds for the restricted external direct product.

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

transitive subgroup property | Yes | direct factor is transitive | If , with a direct factor of and a direct factor of , then is a direct factor of . |

finite-intersection-closed subgroup property | No | direct factor is not finite-intersection-closed | We can have direct factors of but not a direct factor of . |

finite-join-closed subgroup property | No | direct factor is not finite-join-closed | We can have direct factors of but not a direct factor of . |

intermediate subgroup condition | Yes | direct factor satisfies intermediate subgroup condition | If with a direct factor of , then is a direct factor of . |

trim subgroup property | Yes | The whole group and the trivial subgroup are direct factors | |

image condition | No | direct factor does not satisfy image condition | We can have a surjective homomorphism and a direct factor of such that is not a direct factor of . |

quotient-transitive subgroup property | Yes | direct factor is quotient-transitive | If with a direct factor of and a direct factor of , then is a direct factor of . |

upper join-closed subgroup property | No | direct factor is not upper join-closed | We can have and intermediate subgroups such that is a direct factor in each but not in . |

lower central series condition | Yes | direct factor satisfies lower central series condition | Suppose is a direct factor of a group . Then, for any positive integer , the lower central series member is a direct factor of . |

## Relation with other properties

### Stronger properties

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

fully invariant direct factor | direct factor and a fully invariant subgroup | Characteristic direct factor|FULL LIST, MORE INFO | ||

characteristic direct factor | direct factor and a characteristic subgroup | |FULL LIST, MORE INFO | ||

abelian direct factor | direct factor and an abelian group | |FULL LIST, MORE INFO |

### Weaker properties

### Related group properties

Group property | Definition in terms of direct factor |
---|---|

Directly indecomposable group | nontrivial group with no proper nontrivial direct factor |

Complete group | it is a direct factor of any bigger group in which it is a normal subgroup |

Group in which every normal subgroup is a direct factor | every normal subgroup is a direct factor |

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### Monadic second-order description

This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups

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## Effect of property operators

Operator | Meaning | Result of application | Proof |
---|---|---|---|

finite-join-closure | a subgroup that can be expressed as a join of finitely many direct factors. | join of finitely many direct factors | by definition |

join-closure | a subgroup that can be expressed as a join of direct factors. | join of direct factors | by definition |

image-potentially operator | a subgroup that can arise as the image under a surjective homomorphism of a direct factor of some group. | central factor | central factor iff image-potentially direct factor |

## Testing

### GAP code

GAP-codable subgroup propertyOne can write code to test this subgroup property inGAP (Groups, Algorithms and Programming), though there is no direct command for it.

View the GAP code for testing this subgroup property at: IsDirectFactor

View other GAP-codable subgroup properties | View subgroup properties with in-built commands