Left-extensibility-stable implies intermediate subgroup condition

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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 (?))
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Statement

Let p be a left-extensibility-stable subgroup property, viz a subgroup property that can be written as a \to b where a is an extensibility-stable function property.

Then, p satisfies the intermediate subgroup condition, or equivalently, whenever H \le K \le G 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 H \le K \le G be groups such that H satisfies p in G. We need to show that H satisfies p in K.

Let \sigma be a function on K satisfying property p in K. Then, we need to show that \sigma restricts to a function on H which satisfies b in H.

Since a is an extensibility-stable function property, there exists a function \sigma' on G whose restriction to K is \sigma. Now, since H satisfies property a \to b in G, the restriction of \sigma' to H is well-defined and satisfies property b in H.

But the restriction to H of \sigma' is the same as the restriction to H of \sigma. Hence, we have shown that the restriction to H of \sigma is well-defined and satisfies property b over H.