Left-realized 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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This article is about a general term. A list of important particular cases (instances) is available at Category: Left-realized subgroup properties

Definition

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.

Relation with other metaproperties

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