Group property modifier

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.