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Intermediate subgroup condition
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
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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
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.
| Weaker than | Strongly UL-intersection-closed subgroup property +, Inverse image condition +, Transfer condition +, Left-inner subgroup property +, and Left-extensibility-stable subgroup property + |

