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Left-extensibility-stable implies intermediate subgroup condition
From Groupprops
This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., left-extensibility-stable subgroup property) must also satisfy the second subgroup metaproperty (i.e., intermediate subgroup condition)
View all subgroup metaproperty implications | View all subgroup metaproperty non-implications
Statement
Let p be a left-extensibility-stable subgroup property, viz a subgroup property that can be written as 
where a is an extensibility-stable function property.
Then, p satisfies the intermediate subgroup condition, or equivalently, whenever 
with H satisfying p in the whole of G, H also satisfies p in K.
Examples
Some examples of subgroup properties that are left-extensibility-stable and, on account of this, satisfy the intermediate subgroup condition, are:
Proof
Let p be a left-extensibility-stable subgroup property and be groups such that H satisfies p in G. We need to show that H satisfies p in K.
Let σ be a function on K satisfying property p in K. Then, we need to show that σ restricts to a function on H which satisfies b in H.
Since a is an extensibility-stable function property, there exists a function σ' on G whose restriction to K is σ. Now, since H satisfies property in G, the restriction of σ' to H is well-defined and satisfies property b in H.
But the restriction to H of σ' is the same as the restriction to H of σ. Hence, we have shown that the restriction to H of σ is well-defined and satisfies property b over H.
