Intermediate subgroup condition

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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 $H$ ≤ $K$ ≤ $G$ are groups and $H$ satisfies $p$ in $G$, $H$ also satisfies $p$ in $K$.

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

• Subgroup properties satisfying inverse image condition
• Subgroup properties satsfying 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
• Any finite-intersection-closed subgroup property satisfying intermediate subgroup condition must also satisfy transfer condition