# Virtually 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

This is a variation of Abelianness|Find other variations of Abelianness |

## Contents

## Definition

### Symbol-free definition

A group is said to be **virtually Abelian** if it has an Abelian subgroup of finite index.

## Formalisms

### In terms of the virtually operator

This property is obtained by applying the virtually operator to the property: Abelian group

View other properties obtained by applying the virtually operator

## Relation with other properties

### Stronger properties

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

finite group | ||||

abelian group | ||||

FZ-group | center is a subgroup of finite index | FZ implies virtually abelian | ||

locally finite linear group over characteristic zero | ||||

metacyclic group |