Identity-true 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 termed identity-true if it is implied by the improper property or the identity property (for the composition operator), or equivalently, if every group satisfies the property as a subgroup of itself. In other words, a subgroup property $p$ is identity-true if $e$ ≤ $p$.

Definition with symbols

A subgroup property $p$ is termed identity-true if any group $G$ satisfies $p$ as a subgroup of itself.

Property theory

Stability under binary operators

The following are true:

• The composition operator applied to identity-true properties is again an identity-true property.
• The intersection operator applied to identity-true properties is again an identity-true property.
• The subgroup generation operator applied to identity-true properties is again an identity-true property.

Thus, for each of these, the identity-true properties forms a submonoid of the corresponding monoid.

Relation with transitivity

Being identity-true is one of the two conditions for being a t.i. subgroup property, and the latter is one of the most important subgroup metaproperties. The other condition is that of being transitive. The relation and interplay of these properties is captured somewhat in the residuation master theorem and the transiter master theorem.