Virtually operator

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.
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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 $p$ (typically, a subgroup-closed group property) and outputs the property of being a group which has a subgroup of finite index satisfying $p$ 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