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Vipul at 13:28, 6 March 2007

2007-03-06T13:28:05Z

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{{subgroup metaproperty}}

{{wikilocal}}
==How they came about==

==Definition==

===Symbol-free definition===

A subgroup property is termed ''t.i.'' if it is both [[transitive subgroup property|transitive]] and [[identity-true subgroup property|identity-true]] with respect to the [[composition operator]]. That is, <math>p</math> is t.i. if <math>e</math> ≤ <math>p</math> and <math>p * p</math> ≤ <math>p</math>.

===Definition with symbols===

A subgroup property <math>p</math> is termed ''t.i.'' if it satisfies the following two conditions:

* For any group <math>G</math>, <math>G</math> satisfies <math>p</math> as a subgroup of itself. This is the condition of being [[identity-true subgroup property|identity-true]].
* If <math>G</math> ≤ <math>H</math> ≤ <math>K</math>, such that <math>G</math> satisfies <math>p</math> in <math>H</math> and <math>H</math> satisfies <math>p</math> in <math>K</math> then <math>G</math> satisfies <math>p</math> in <math>K</math>. This is the condition of being [[transitive subgroup property|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 <math>p</math>. Then the map sending an arbitrary property <math>q</math> to the conjunction of <math>p</math> with <math>q</math>, is an endomorphism of the property monoid if and only if <math>p</math> 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 operator]]s :

* The [[left transiter]] operator
* The [[right transiter]] operator
* The [[subordination operator]]

T.i. subgroup property - Revision history

Vipul: 1 revision

Vipul at 13:28, 6 March 2007