# Left-realized subgroup property

From Groupprops

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

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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 is said to be **left-realized** if, given any group , there is a group containing such that satisfies the property as a subgroup of .

### 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

### Opposite

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