# Virtually operator

From Groupprops

*This article defines a group property modifier (a unary group property operator) -- viz an operator that takes as input a group property and outputs a group property*

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Virtually operator, all facts related to Virtually operator) |Survey articles about this | Survey articles about definitions built on this

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## Definition

### Symbol-free definition

The **virtually operator** is a operator from the group property space to itself that takes as input a group property (typically, a subgroup-closed group property) and outputs the property of being a group which has a subgroup of finite index satisfying as an abstract group.

## Application

Inportant instances of application of the virtually operator:

- virtually nilpotent group: obtained from nilpotent group
- virtually Abelian group: obtained from Abelian group