Normality satisfies intermediate subgroup condition

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This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement

Verbal statement

If a subgroup is normal in the whole group, it is also normal in every intermediate subgroup of the group containing it.

Statement with symbols

Let $H\le K\le G$ be groups such that $H◃G$ (viz $H$ is normal in $G$). Then, $H$ is normal in $K$.

Property-theoretic statement

The subgroup property of being normal satisfies the Intermediate subgroup condition (?).

Generalizations

Stronger metaproperties satisfied by normality

• Normality satisfies transfer condition
• Normality satisfies inverse image condition

Weaker conditions to ensure intermediate subgroup condition

• Left-inner implies intermediate subgroup condition
• Left-extensibility-stable implies intermediate subgroup condition

Related results

Other subgroup properties satisfying intermediate subgroup condition

• Central factor satisfies intermediate subgroup condition
• Direct factor satisfies intermediate subgroup condition

Proof

Hands-on proof

Given $H\le K\le G$ such that $H◃G$, we need to show that $H◃K$. To prove this, it suffices to show that for any $g\in K$, $gH{g}^{-1}=H$.

Pick any $g\in K$. Then, since $K\le G$, $g\in G$. But since $H$ is normal in $G$, $gH{g}^{-1}=H$. This proves it.

Deeper insight leading to generalization

We need to show that given any inner automorphism $\sigma$ of $K$, $H$ is invariant under $\sigma$.

We know that given any inner automorphism of the whole group $G$, $H$ is invariant under that. Thus, what we need to do is extend $\sigma$ to an inner automorphism ${\sigma }^{\prime }$ of the whole of $G$. In other words, we need to show that any inner automorphism of a subgroup can be lifted to an inner automorphism of the whole group.

This in turn follows easily from the fact that an inner automorphism is described via conjugation by an element of the subgroup, and conjugation by the same element also defines an inner automorphism on the whole group.

This leads to the generalizations mentioned above: any left-inner subgroup property satisfies the intermediate subgroup condition and any left-extensibility-stable subgroup property satisfies the intermediate subgroup condition.