Left-extensibility-stable implies intermediate subgroup condition

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 must also satisfy the second subgroup metaproperty
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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:

• Normal subgroup
• Central factor

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 ${\sigma }^{\prime }$ on $G$ whose restriction to $K$ is $\sigma$. Now, since $H$ satisfies property $a\to b$ in $G$, the restriction of ${\sigma }^{\prime }$ to $H$ is well-defined and satisfies property $b$ in $H$.

But the restriction to $H$ of ${\sigma }^{\prime }$ 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 $H$.