Potentially operator

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
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 $$p \le q$$ implies that potentially $$p \le q$$.

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

Any subgroup which satisfies property $$p$$ must also satisfy potentially $$p$$.

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.