Left-realized subgroup property

Symbol-free definition
A subgroup property is said to be left-realized if every group can be realized as a subgroup having that property (inside some group).

Definition with symbols
A subgroup property $$p$$ is said to be left-realized if, given any group $$G$$, there is a group $$H$$ containing $$G$$ such that $$G$$ satisfies the property $$p$$ as a subgroup of $$H$$.

In terms of realization operators
A subgroup property is said to be left-realized if applying the left realization operator to it gives the tautological group property.

Stronger metaproperties

 * Identity-true subgroup property
 * Trim subgroup property
 * t.i. subgroup property

Opposite
A subgroup property which is not left-realized is termed left-unrealized.

Related properties

 * Right-realized subgroup property