# 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 $H$.