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 $H \le K \le G$ 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 $H \le G$ 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.

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

Property	Meaning	Proof that it satisfies intermediate subgroup condition	Stronger metaproperties it satisfies
normal subgroup	invariant under all inner automorphisms	normality satisfies intermediate subgroup condition	left-inner subgroup property, left-extensibility-stable subgroup property, strongly UL-intersection-closed subgroup property
central factor	every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.	central factor satisfies intermediate subgroup condition	left-inner subgroup property
direct factor	factor in an internal direct product	direct factor satisfies intermediate subgroup condition
complemented normal subgroup	normal subgroup with a permutable complement.	complemented normal satisfies intermediate subgroup condition
pronormal subgroup	any conjugate to it is conjugate in their join.	pronormality satisfies intermediate subgroup condition
subnormal subgroup	series from subgroup to whole group, each normal in the next.	subnormality satisfies intermediate subgroup condition
isomorph-containing subgroup	contains any subgroup isomorphic to itself.
homomorph-containing subgroup	contains any homomorphic image of itself.

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 $H \le G$ satisfies the property, and $K \le G$, then $H \cap K$ satisfies the property in $K$.
left-inner subgroup property	any subgroup property that can be expressed using a function restriction expression of the form inner $\to$ 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

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