Intermediate subgroup condition: Difference between revisions

Revision as of 23:14, 19 February 2013

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

A subgroup property $p$ is said to satisfy the intermediate subgroup condition if whenever $H\leq K\leq G$ are groups and $H$ satisfies $p$ in $G$ , $H$ also satisfies $p$ in $K$ .

Formalisms

This article defines a single-input-expressible subgroup metaproperty

Consider a procedure $P$ that takes as input a group-subgroup pair $H\leq G$ and outputs all group-subgroup pairs $H\leq K$ where $K$ is an intermediate subgroup of $G$ containing $H$ . Then, the intermediate subgroup condition is the single-input-expressible subgroup property corresponding to procedure $P$ . In other words, a subgroup property $p$ satisfies the intermediate subgroup condition if whenever $H\leq G$ satisfies property $p$ , all the pairs obtained by applying procedure $P$ to $H\leq G$ also satisfy property $p$ .

In terms of the intermediately operator

A subgroup property satisfies intermediate subgroup condition if and only if it is a fixed-point of the idempotent subgroup property modifier called the intermediately operator.

In terms of the potentially operator

A subgroup property satisfies intermediate subgroup condition if and only if it is a fixed-point of the idempotent subgroup property modifier called the potentially operator.

Metametaproperties

Metametaproperty name	Satisfied?	Proof	Statement with symbols
conjunction-closed subgroup metaproperty	Yes	follows from being single-input-expressible.	A conjunction (AND) of subgroup properties, each satisfying the intermediate subgroup condition, also satisfies the intermediate subgroup condition.
disjunction-closed subgroup metaproperty	Yes	follows from being single-input-expressible.	A disjunction (OR) of subgroup properties, each satisfying the intermediate subgroup condition, also satisfies the intermediate subgroup condition.
right residual-preserved subgroup metaproperty	Yes		The right residual of a subgroup property satisfying the intermediate subgroup condition, by any subgroup property, is a subgroup property satisfying the intermediate subgroup condition.

Relation with other metaproperties

Stronger metaproperties

Metaproperty name	Meaning	Proof of implication
Strongly UL-intersection-closed subgroup property
inverse image condition	inverse image of a subgroup satisfying the property under any homomorphism of groups satisfies the property.
transfer condition	If $H\leq G$ satisfies the property, and $K\leq G$ , then $H\cap K$ satisfies the property in $K$ .
left-inner subgroup property	any subgroup property that can be expressed using a function restriction expression of the form inner $\to$ something, i.e., every inner automorphism of the whole group restricts to a function of the subgroup satisfying some conditions purely in terms of the subgroup.	(via left-extensibility-stable)
left-extensibility-stable subgroup property		left-extensibility-stable implies intermediate subgroup condition

Weaker metaproperties

Conjunction implications

Any left-realized subgroup property satisfying intermediate subgroup condition must be identity-true. For full proof, refer: Left-realized and intermediate subgroup condition implies identity-true

@@ Line 20: / Line 20: @@
 A subgroup property satisfies intermediate subgroup condition if and only if it is a fixed-point of the idempotent [[subgroup property modifier]] called the [[potentially operator]].
+==Metametaproperties==
+{| class="sortable" border="1"
+! Metametaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metametaproperty::conjunction-closed subgroup metaproperty]] || Yes || follows from being single-input-expressible. || A conjunction (''AND'') of subgroup properties, each satisfying the intermediate subgroup condition, also satisfies the intermediate subgroup condition.
+|-
+| [[satisfies metametaproperty::disjunction-closed subgroup metaproperty]] || Yes || follows from being single-input-expressible. || A disjunction (''OR'') of subgroup properties, each satisfying the intermediate subgroup condition, also satisfies the intermediate subgroup condition.
+|-
+| [[satisfies metametaproperty::right residual-preserved subgroup metaproperty]] || Yes || || The [[right residual]] of a subgroup property satisfying the intermediate subgroup condition, by any subgroup property, is a subgroup property satisfying the intermediate subgroup condition.
+|}
 ==Relation with other metaproperties==
@@ Line 25: / Line 37: @@
 ===Stronger metaproperties===
-* [[Weaker than::Strongly UL-intersection-closed subgroup property]]
+{| class="sortable" border="1"
-* [[Weaker than::inverse image condition]]
+! Metaproperty name !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediat enotions
-* [[Weaker than::transfer condition]]
+|-
-* [[weaker than::Left-inner subgroup property]]
+| [[Weaker than::Strongly UL-intersection-closed subgroup property]] || || || ||
-* [[weaker than::Left-extensibility-stable subgroup property]]: {{proofat|[[Left-extensibility-stable implies intermediate subgroup condition]]}}
+|-
+| [[Weaker than::inverse image condition]] || inverse image of a subgroup satisfying the property under any homomorphism of groups satisfies the property. || || ||
+|-
+| [[Weaker than::transfer condition]] || If <math>H \le G</math> satisfies the property, and <math>K \le G</math>, then <math>H \cap K</math> satisfies the property in <math>K</math>. || || ||
+|-
+| [[weaker than::left-inner subgroup property]] || any subgroup property that can be expressed using a [[function restriction expression]] of the form inner <math>\to</math> something, i.e., every inner automorphism of the whole group restricts to a function of the subgroup satisfying some conditions purely in terms of the subgroup. || (via left-extensibility-stable) || ||
+|-
+| [[weaker than::left-extensibility-stable subgroup property]] || || [[left-extensibility-stable implies intermediate subgroup condition]] || ||
+|}
 ===Weaker metaproperties===
@@ Line 36: / Line 56: @@
 * Any [[left-realized subgroup property]] satisfying intermediate subgroup condition must be [[identity-true subgroup property|identity-true]]. {{proofat|[[Left-realized and intermediate subgroup condition implies identity-true]]}}
-==Metametaproperties==
-{{conjunction-closed subgroup metaproperty}}
-A conjunction (''AND'') of subgroup properties, each satisfying the intermediate subgroup condition, also satisfies the intermediate subgroup condition. This follows from the fact that it is a single-input-expressible subgroup metaproperty
-{{disjunction-closed subgroup metaproperty}}
-A disjunction (''OR'') of subgroup properties, each satisfying the intermediate subgroup condition, also satisfies the intermediate subgroup condition. This follows from the fact that it is a single-input-expressible subgroup metaproperty
-{{right residual-preserved subgroup metaproperty}}
-The right residual of a subgroup property satisfying the intermediate subgroup condition, by any subgroup property, is a subgroup property satisfying the intermediate subgroup condition.