Intermediate subgroup condition
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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Definition
A subgroup property is said to satisfy the intermediate subgroup condition if whenever are groups and satisfies in , also satisfies in .
Formalisms
This article defines a single-input-expressible subgroup metaproperty
Consider a procedure that takes as input a group-subgroup pair and outputs all group-subgroup pairs where is an intermediate subgroup of containing . Then, the intermediate subgroup condition is the single-input-expressible subgroup property corresponding to procedure . In other words, a subgroup property satisfies the intermediate subgroup condition if whenever satisfies property , all the pairs obtained by applying procedure to also satisfy property .
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.
Examples
For more on how to prove that a subgroup property satisfies this, see proving intermediate subgroup condition.
Examples of important subgroup properties satisfying this
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 | Proof of strictness (reverse implication failure) | Intermediat enotions |
---|---|---|---|---|
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 satisfies the property, and , then satisfies the property in . | |||
left-inner subgroup property | any subgroup property that can be expressed using a function restriction expression of the form inner 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