Normality satisfies transfer condition

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

Verbal statement

If a subgroup is normal in the group, its intersection with any other subgroup is normal in that subgroup.

Symbolic statement

Let $H\le G$ be a normal subgroup and let $K$ be any subgroup of $G$. Then, $H\cap K◃K$.

Property-theoretic statement

The subgroup property of being normal satisfies the transfer condition.

Definitions used

Normal subgroup

A subgroup $H$ of a group $G$ is said to be normal if for any $g\in G$ and $h\in H$, $gh{g}^{-1}\in H$.

Transfer condition

A subgroup property $p$ is said to satisfy transfer condition if whenever $H,K$ are subgroups of $G$ and $H$ has property $p$ in $G$, $H\cap K$ has property $p$ in $K$.

Generalizations

Stronger metaproperties satisfied by normality

• Normality satisfies inverse image condition

Proof

Hands-on proof

Suppose $H◃G$ and $K\le G$. We need to prove that $H\cap K◃K$. In other words, we need to prove that given any $g\in K$ and $h\in H\cap K$, $gh{g}^{-1}\in H\cap K$.

Here's how the proof proceeds. Since $h\in H\cap K$, we in particular have $h\in H$. Since $H◃G$ (viz $H$ is normal in $G$), $gh{g}^{-1}\in H$.

But we also have that $g\in K$ and $h\in K$. Since $K$ is a subgroup, $gh{g}^{-1}\in K$.

Combining these two facts, $gh{g}^{-1}\in H\cap K$.