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 metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to T.i. subgroup property, all facts related to T.i. subgroup property) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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, $p$ is t.i. if $e$ ≤ $p$ and $p*p$ ≤ $p$.

Definition with symbols

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

• For any group $G$, $G$ satisfies $p$ as a subgroup of itself. This is the condition of being identity-true.
• If $G$ ≤ $H$ ≤ $K$, such that $G$ satisfies $p$ in $H$ and $H$ satisfies $p$ in $K$ then $G$ satisfies $p$ in $K$. 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 $p$. Then the map sending an arbitrary property $q$ to the conjunction of $p$ with $q$, is an endomorphism of the property monoid if and only if $p$ 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