# Perfect group

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## Definition

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

No. | Shorthand | A group is said to be perfect if ... | A group is said to be perfect if ... |
---|---|---|---|

1 | equals own derived subgroup | it equals its own derived subgroup (i.e., its commutator subgroup with itself). | equals the derived subgroup , sometimes also denoted . |

2 | every element is a product of commutators | every element of the group can be expressed as a product in the group of finitely many elements each of which is a commutator. | for any , there exist elements (with possible repetitions) such that . |

3 | trivial abelianization | its abelianization is a trivial group. | the abelianization is the trivial group. |

4 | trivial homomorphism to any abelian group | any homomorphism of groups from it to an abelian group is the trivial homomorphism. | for any abelian group , and any homomorphism of groups , must send all elements of to the identity element of . |

This definition is presented using a tabular format. |View all pages with definitions in tabular format

### In terms of the fixed-point operator

The property of being perfect is obtained by applying the fixed-point operator to a subgroup-defining function, namely the derived subgroup.

## Examples

### Extreme examples

- The trivial group is a perfect group.

### Groups satisfying the property

Here are some basic/important groups satisfying the property:

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

Trivial group | 1 (1) |

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

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

Alternating group:A5 | 60 (5) |

Alternating group:A6 | 360 (118) |

Projective special linear group:PSL(3,2) | 168 (42) |

Special linear group:SL(2,5) | 120 (5) |

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

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

Alternating group:A7 | |

Projective special linear group:PSL(2,11) | 660 (13) |

Projective special linear group:PSL(2,8) | 504 (156) |

Special linear group:SL(2,7) | 336 (114) |

Special linear group:SL(2,9) | 720 (409) |

### Groups dissatisfying the property

Note that any nontrivial solvable group cannot be a perfect group, so this gives lots of non-examples. The examples discussed below concentrate more on the non-solvable groups that still fail to be perfect.

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 | |
---|---|

Symmetric group:S5 | 120 (34) |

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

## Metaproperties

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

subgroup-closed group property | No | perfectness is not subgroup-closed | It is possible to have a perfect group and a subgroup of such that is not perfect. In fact, any nontrivial perfect group has a nontrivial cyclic subgroup that is not perfect. (in fact, every finite group is a subgroup of a finite perfect group). |

characteristic subgroup-closed group property | No | perfectness is not characteristic subgroup-closed | It is possible to have a perfect group and a characteristic subgroup of such that is not perfect. |

quotient-closed group property | Yes | perfectness is quotient-closed | If is a perfect group and is a normal subgroup of , the quotient group is also a perfect group. |

finite direct product-closed group property | Yes | perfectness is finite direct product-closed | Suppose are all (possibly isomorphic, possibly non-isomorphic) perfect groups. Then, the external direct product is also a perfect group. |

join-closed group property | Yes | perfectness is join-closed | Suppose is a (possibly finite, possibly infinite) collection of subgroups of a group , such that each is a perfect group. Then, the join of subgroups is a perfect group. |

## Relation with other properties

### Stronger properties

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

superperfect group | perfect and Schur-trivial: the Schur multiplier is trivial. | (by definition) | follows from perfect not implies Schur-trivial | |FULL LIST, MORE INFO |

simple non-abelian group | non-abelian and has no proper nontrivial normal subgroup. | simple and non-abelian implies perfect | Quasisimple group, Semisimple group|FULL LIST, MORE INFO | |

quasisimple group | perfect and inner automorphism group is a simple non-abelian group | Semisimple group|FULL LIST, MORE INFO | ||

group in which every element is a commutator | every element of the group is a commutator of two elements of the group. | |FULL LIST, MORE INFO |

### Weaker properties

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

square-in-derived group | every square element is in the derived subgroup. | |FULL LIST, MORE INFO | ||

stem group | the center is contained in the derived subgroup | |FULL LIST, MORE INFO | ||

group with unique Schur covering group | the Schur covering group is unique, or equivalently, of the abelianization over the Schur multiplier is trivial. | |FULL LIST, MORE INFO |

## Testing

### GAP command

This group property can be tested using built-in functionality ofGroups, Algorithms, Programming(GAP).

The GAP command for this group property is:IsPerfectGroup

View GAP-testable group properties

To test whether a given group is perfect, the command is:

IsPerfectGroup(group);

The command:

PerfectGroup(n,r)

gives the perfect group of order . If is not specified, this simply gives the first perfect group of order . An error is thrown if there are no perfect groups of order .

## References

### Textbook references

Book | Page number | Chapter and section | Contextual information | View |
---|---|---|---|---|

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info} |
174 | definition introduced in exercise (Problem 19) | ||

A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info} |
157 | Section 5.4, Problem 4 | definition introduced in exercise | Google Books |

Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754^{More info} |
27 | Google Books |

## External links

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