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Intermediate subgroup condition

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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


Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 H \le K \le G 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 H \le G and outputs all group-subgroup pairs H \le 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 \le G satisfies property p, all the pairs obtained by applying procedure P to H \le 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.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties

Conjunction implications

Metametaproperties

Conjunction-closedness

This subgroup metaproperty is conjunction-closed: an arbitrary conjunction (AND) of subgroup properties satisfying this metaproperty, also satisfies this metaproperty
View 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 all 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 right 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.

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