Group property modifier

This article is about the notion of property modifier for the [[{{{1}}} property space]]: it inputs a [[{{{1}}} property]] and outputs a [[{{{1}}} property]]

This article is about a general term. A list of important particular cases (instances) is available at Category:Group property modifiers

Definition

Symbol-free definition

A group property modifier is a function from the group property space to itself that takes as input a group property and outputs a subgroup property.

Examples

An example is the locally operator. The locally operator takes as input a group property $p$, and outputs a new property: the property of being a group in which every finitely generated subgroup has property $p$. Thus, the locally operator modifies $p$ to give a new property.

Related notions

Given a group property modifier, we are often interested in its image space: the collection of group properties that can be obtained by applying this modifier to some group property. We are also interested in the fixed-point space: the collection of group properties that are unaffected by the modifier. An idempotent group property modifier is a group property modifier whose fixed-point space coincides with its image space; a number of group property modifiers occurring naturally are idempotent.