Potentially operator

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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 property

View 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 $p$, the subgroup property potentially $p$ 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 $p$.

Definition with symbols

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

$H$ has property potentially $p$ in $G$ if there exists a group $K$ containing $G$ such that $H$ has property $p$ in $K$.

Properties

Monotonicity

This subgroup property modifier is monotone, viz if $p\leq q$ are subgroup properties and $f$ is the operator, then $f(p)\leq f(q)$

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

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

Also, if $p$ is potential-tautological, then $q$ 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 $p$ must also satisfy potentially $p$.

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

• Strongly potentially operator
• Intermediately operator

Properties obtained via this operator

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