Left-realized subgroup property

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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 metaproperty
VIEW 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


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


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

Related properties