# Potentially operator

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup propertyView a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

*This property modifier is idempotent and a property is a fixed-point, or equivalently, an image of this if and only if it is a:*intermediate subgroup condition

## History

*This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page*

## Definition

### Symbol-free definition

The **potentially operator** is a map from the subgroup property space to itself defined as follows. Given a subgroup property , the subgroup property *potentially* is defined as the property of being a subgroup in a group such that there exists some group containingthe bigger group, in which the subgroup has property .

### Definition with symbols

The **potentially operator** is a map from the subgroup property space to itself defined as follows. Given a subgroup property , the subgroup property *potentially* is defined as follows:

has property *potentially* in if there exists a group containing such that has property in .

## Properties

### Monotonicity

*This subgroup property modifier is monotone, viz if are subgroup properties and is the operator, then *

If every subgroup satisfying property also satisfies , then every subgroup satisfying potentially must also satisfy potentially .

In particular, if satisfies the intermediate subgroup condition (and is hence invariant under the potentially operator), then implies that potentially .

Also, if is potential-tautological, then must also be potential-tautological.

### Ascendance

*This subgroup property modifier is ascendant, viz the image of any subgroup property under this modifier is always weaker than the subgroup property we started with*

Any subgroup which satisfies property must also satisfy potentially .

### Idempotence

*This subgroup property modifier is idempotent, viz applying it twice to a subgroup property has the same effect as applying it once*

The potentially operator is idempotent, and the subgroup properties that are invariant under this operator are precisely the subgroup properties that satisfy the intermediate subgroup condition.

## Related operators

## Effect on metaproperties

## Properties obtained via this operator

The most important of properties obtained dircetly via this operator is the property of being a potentially characteristic subgroup.

- Terminology local to the wiki
- Nonstandard terminology
- Subgroup property modifiers
- Subgroup property operators
- Idempotent operators that fix properties satisfying the following: intermediate subgroup condition
- Monotone subgroup property modifiers
- Ascendant subgroup property modifiers
- Idempotent subgroup property modifiers