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From Groupprops

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

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Formalisms 2.1 In terms of the intermediately operator 2.2 In terms of the potentially operator 3 Relation with other metaproperties 3.1 Stronger metaproperties 3.2 Weaker metaproperties 3.3 Conjunction implications 4 Metametaproperties 4.1 Conjunction-closedness 4.2 Disjunction-closedness 4.3 Effect of right residuals

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This article is about a general term. A list of important particular cases (instances) is available at Category: Subgroup properties satisfying intermediate subgroup condition

Definition

Symbol-free definition

A subgroup property p is said to satisfy the intermediate subgroup condition if whenever a subgroup satisfies property p in the whole group, it also satisfies p as a subgroup of every intermediate subgroup.

Definition with symbols

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

Formalisms

Template:Singleinput

Consider a procedure P that takes as input a group-subgroup pair and outputs all group-subgroup pairs 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 satisfies property p, all the pairs obtained by applying procedure P to 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 operator 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 operator called the potentially operator.

Relation with other metaproperties

Stronger metaproperties

• Strongly UL-intersection-closed subgroup property
• inverse image condition
• transfer condition
• Left-inner subgroup property
• Left-extensibility-stable subgroup property: For full proof, refer: 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

Metametaproperties

Conjunction-closedness

This subgroup metaproperty is conjunction-closed: an arbitrary conjunction (AND) of subgroup properties satisfying this metaproperty, also satisfies this metaproperty
View a complete list of conjunction-closed subgroup metaproperties

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-closedness

This subgroup metaproperty is disjunction-closed: an arbitrary disjunction (OR) of subgroup properties satisfying this metaproperty, also satisfies this metaproperty
View a complete list of disjunction-closed subgroup metaproperties

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

Effect of right residuals

This subgroup metaproperty is right residual-preserved: the left residual of any subgroup property satisfying this metaproperty, by any subgroup property, also satisfies this metaproperty.
View a complete list of such metaproperties

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.