# Identity-true subgroup property

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]

## How they came about

## 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 is identity-true if ≤ .

### Definition with symbols

A subgroup property is termed **identity-true** if any group satisfies 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.