# Complete group

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 Complete group, all facts related to Complete group) |Survey articles about this | Survey articles about definitions built on this

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

### Symbol-free definition

A group is said to be **complete** if it satisfies the following equivalent conditions:

- It is centerless and every automorphism of it is inner.
- The natural homomorphism to the automorphism group, that sends each element to the conjugation via that element, is an isomorphism.
- Whenever it is embedded as a normal subgroup inside a bigger group, it is actually a direct factor inside that bigger group.

### Definition with symbols

A group is said to be **complete** if it satisfies the following equivalent conditions:

- (the center of ) is trivial and (i.e., every automorphism of is inner)
- The natural homomorphism given by (where ) is an isomorphism
- For any embedding of as a normal subgroup of some group , is a direct factor of

## Examples

### Extreme examples

- The trivial group is complete.

### Groups satisfying the property

Here are some basic/important groups satisfying the property:

GAP ID | |
---|---|

Symmetric group:S3 | 6 (1) |

Here are some relatively less basic/important groups satisfying the property:

GAP ID | |
---|---|

Symmetric group:S4 | 24 (12) |

Symmetric group:S5 | 120 (34) |

Here are some even more complicated/less basic groups satisfying the property:

### Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:

Here are some relatively less basic/important groups that do not satisfy the property:

GAP ID | |
---|---|

Alternating group:A5 | 60 (5) |

Alternating group:A6 | 360 (118) |

Here are some even more complicated/less basic groups that do not satisfy the property:

GAP ID | |
---|---|

Alternating group:A7 | |

M16 | 16 (6) |

## Formalisms

### In terms of the supergroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties in the following sense. Whenever the given group is embedded as a subgroup satisfying the first subgroup property (normal subgroup), in some bigger group, it also satisfies the second subgroup property (direct factor), and vice versa.

View other group properties obtained in this way

A group is complete if and only if whenever is embedded as a normal subgroup in some group , is a direct factor of .

## Relation with other properties

### Stronger properties

- Symmetric group on a set of size other than or :
`For full proof, refer: Symmetric groups on finite sets are complete, Symmetric groups on infinite sets are complete` - Automorphism group of a non-Abelian characteristically simple group:
`For full proof, refer: Characteristically simple and non-Abelian implies automorphism group is complete`

### Weaker properties

- Group in which every automorphism is inner
- Group in which every normal subgroup is characteristic
- Centerless group
- Group isomorphic to its automorphism group

## Testing

### GAP code

One can write code to test this group property inGAP (Groups, Algorithms and Programming), though there is no direct command for it.

View the GAP code for testing this group property at: IsCompleteGroup

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

While there is no built-in command to test completeness, this can be done with a short snippet of code available at GAP:IsCompleteGroup. The function is invoked as follows:

IsCompleteGroup(group);