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
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VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions
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How they came about
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.
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.
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 :