# Schur-trivial group

From Groupprops

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

## Contents

## Definition

A group is said to be **Schur-trivial** or a **group with trivial Schur multiplier** if it satisfies the following equivalent conditions:

- Its Schur multiplier is the trivial group.
- The homomorphism from its exterior square to its derived subgroup defined by the commutator map is an isomorphism of groups to the derived subgroup.
- It is isomorphic to its own Schur covering group with the covering map being the identity map.

## Examples

### Extreme examples

- The trivial group is Schur-trivial.
- Cyclic groups are Schur-trivial.

### Groups satisfying the property

Here are some basic/important groups satisfying the property:

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

Cyclic group:Z2 | 2 (1) |

Symmetric group:S3 | 6 (1) |

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

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

Quaternion group | 8 (4) |

Special linear group:SL(2,3) | 24 (3) |

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

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

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

Generalized quaternion group:Q16 | 16 (9) |

M16 | 16 (6) |

Semidihedral group:SD16 | 16 (8) |

### 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) |

Dihedral group:D8 | 8 (3) |

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 | Schur-triviality is not subgroup-closed | It is possible to have a Schur-trivial group and a subgroup of such that is not Schur-trivial. (For a finite group, every subgroup being Schur-trivial is equivalent to the group being a finite group with periodic cohomology). |

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

quotient-closed group property | No | Schur-triviality is not quotient-closed | It is possible to have a Schur-trivial group and a normal subgroup of such that the quotient group is not Schur-trivial. |

finite direct product-closed group property | No | Schur-triviality is not finite direct product-closed | It is possible to have two Schur-trivial groups and such that the external direct product is not Schur-trivial. |

isoclinism-invariant group property | No | Schur-triviality is not isoclinism-invariant | It is possible to have isoclinic groups and such that is Schur-trivial but is not Schur-trivial. |

## Facts

- Finite group generated by Schur-trivial subgroups of relatively prime indices is Schur-trivial
- All Sylow subgroups are Schur-trivial implies Schur-trivial

## Relation with other properties

### Stronger properties

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

cyclic group | generated by one element | Schur multiplier of cyclic group is trivial | the smallest non-cyclic group with trivial Schur multiplier is symmetric group:S3. | |FULL LIST, MORE INFO |

free group | has a freely generating set | Schur multiplier of free group is trivial | any finite cyclic group gives a counterexample. | |FULL LIST, MORE INFO |

Z-group | finite group in which every Sylow subgroup is cyclic | Z-group implies Schur-trivial | (infinite free groups; also, lots of counterexamples among finite groups, such as the quaternion group and various Schur covering groups of centerless groups. For instance, SL(2,3), SL(2,5)) | Finite group with periodic cohomology|FULL LIST, MORE INFO |

finite group with periodic cohomology | finite group in which every abelian subgroup is cyclic. Equivalently, every Sylow subgroup for odd primes is cyclic, and the 2-Sylow subgroup is cyclic or a generalized quaternion group | finite group with periodic cohomology is Schur-trivial | |FULL LIST, MORE INFO | |

superperfect group | perfect and Schur-trivial | |FULL LIST, MORE INFO |

### Weaker properties

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

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