Intermediate subgroup condition

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.

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

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
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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.
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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.