# Group with unique Schur covering 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

## Statement

A **group with unique Schur covering group** is a group satisfying the following equivalent conditions:

- It has a unique Schur covering group, i.e., given Schur covering groups of with covering maps and , there is an isomorphism such that .
- where is the abelianization of (and also identified with ) while is the Schur multiplier of (and also identified with ). Note that for a finite group, this is equivalent to verifying that and have relatively prime orders.

## Examples

The smallest example of a group where neither the abelianization nor the Schur multiplier is trivial, but the Schur covering group is still unique, is alternating group:A4, whose unique Schur multiplier is special linear group:SL(2,3).

In addition, we can take perfect groups and Schur-trivial groups as examples.

### Non-examples

The smallest non-example is the Klein four-group, which has two Schur covering groups (dihedral group:D8 and quaternion group). Note that in general the number of Schur covering groups is .

## Relation with other properties

### Stronger properties

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

perfect group | the abelianization is trivial. | (any Schur-trivial group that is not perfect -- such as the quaternion group) or (alternating group:A4) gives a counterexample | |FULL LIST, MORE INFO | |

Schur-trivial group | the Schur multiplier is trivial | (any perfect group that is not Schur-trivial -- such as alternating group:A5) or (alternating group:A4) gives a counterexample | |FULL LIST, MORE INFO | |

cyclic group | (via Schur-trivial) | (via Schur-trivial) | ||

simple non-abelian group | (via perfect) | (via perfect) |