# Finite group admitting a bijective quasihomomorphism to an abelian 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 finite group is termed a **finite group admitting a bijective quasihomomorphism to an abelian group** if there is an abelian group (in particular, a finite abelian group) and a bijective function that is a quasihomomorphism of groups: whenever commute, we have:

Equivalently, it is a finite nilpotent group and each of its Sylow subgroups is a group of prime power order admitting a bijective quasihomomorphism to an abelian group.

## Relation with other properties

### Stronger properties

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

Finite Lazard Lie group | ||||

Finite abelian group | ||||

Group of prime power order admitting a bijective quasihomomorphism to an abelian group |

### Weaker properties

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

Finite group that is 1-isomorphic to an abelian group | ||||

Finite group that is order statistics-equivalent to an abelian group |