Potential-tautological subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

Symbol-free definition

A subgroup property is said to be potential-tautological if its image under the potentially operator is the tautology subgroup property viz the property of being any subgroup.

Definition with symbols

A subgroup property $p$ is said to be potential-tautological if whenever $H\leq G$ is a subgroup, we can find a group $K$ containing $G$ such that $H$ satisfies property $p$ in $K$.

Relation with other metaproperties

Weaker metaproperties

• Potential-nontrivial-tautological subgroup property: Every nontrivial subgroup has property potentially $p$.
• Potential-proper-tautological subgroup property: Every proper subgroup has property potentially $p$.