# T.i. 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 *t.i.* if it is both transitive and identity-true with respect to the composition operator. That is, is t.i. if ≤ and ≤ .

### Definition with symbols

A subgroup property is termed *t.i.* if it satisfies the following two conditions:

- For any group , satisfies as a subgroup of itself. This is the condition of being identity-true.
- If ≤ ≤ , such that satisfies in and satisfies in then satisfies in . This is the condition of being transitive.

## Property theory

### Property submonoid

The *natural significance* of t.i. properties with respect to the composition operator arises as follows. Consider the property space of all subgroup properties, equipped with a monoid structure via the composition operator. Now take any subgroup property . Then the map sending an arbitrary property to the conjunction of with , is an endomorphism of the property monoid if and only if is a t.i. subgroup property.

- The identity-trueness is needed to ensure that the identity element is preserved.
- The transitivity is needed to ensure that the multiplicative structure is preserved.

Thus, conjunction with a t.i. subgroup property gives a property submonoid.

### Category-theoretic interpretation

If we consider the category whose objects are groups and whose morphisms are injective group homomorphisms, then t.i. subgroup properties are precisely the properties that describe *subcategories* of this category.

### Fixed point space of idempotent operators

The collection of t.i. subgroup properties is precisely the fixed point space of the following three idempotent subgroup operators :

- The left transiter operator
- The right transiter operator
- The subordination operator