Left transiter of normal is characteristic

Statement

Symbolic statement

Let $H\le K$ be a subgroup. The following are equivalent:

1. $H$ is a characteristic subgroup of $K$
2. If $G$ is a group containing $K$ such that $K$ is a normal subgroup of $G$, then $H$ is also a normal subgroup of $G$.

Property-theoretic statement

The left transiter of the subgroup property of normality is the subgroup property of characteristicity. In other words:

• Characteristic $*$ Normal $\le$ Normal

Every characteristic subgroup of a normal subgroup is normal (the $*$ here is for the composition operator).

• If $p$ is such that:

$p*$ Normal $\le$ Normal

Then $p\le$ Characteristic

Proof

1 implies 2

For full proof, refer: Characteristic of normal implies normal

To prove that $H$ being characteristic in $K$ implies the second condition, we need to show that if we start off with any inner automorphism of $G$, it leaves $H$ invariant. We do this by restricting, first to $K$, and then from $K$ to $H$.

2 implies 1

We first sketch a hands-on proof, and then discuss the more general idea.

For the group $K$, let $Hol(K)$ denote the holomorph of $K$. $Hol(K)$ is the semidirect product of $K$ with $Aut(K)$. Observe that every automorphism of $K$ lifts to an inner automorphism of $Hol(K)$, and further, that $K$ is a normal subgroup of $Hol(K)$.

Now let $H$ be a subgroup of $K$ with the property that whenever $K$ is normal in a group $G$, so is $H$. We will s how that $H$ is characteristic in $K$.

Take $G=Hol(K)$. Clearly $K$ is normal in $G$. Now, let $\sigma \in Aut(K)$. Clearly, there is an inner automorphism of $G$ whose restriction to $K$ is $\sigma$. But since every inner automorphism of $G$ must leave $H$ invariant, and hence $\sigma$ must leave $H$ invariant. This shows that $H$ is invariant under all automorphisms of $K$, and hence, is a characteristic subgroup.

Property-theoretic proof

To understand the proof property-theoretically, let us look at the function restriction expression for normality:

Inner automorphism $\to$ Automorphism

It turns out that this function restriction expression for normality is right tight. In other words, we cannot replace Automorphism on the right by any stronger property. Equivalently, every automorphism can be realized as the restriction of an inner automorphism of a bigger group, for some embedding as a normal subgroup.

(In fact, that is exactly what the holomorph construction above shows).

Now, the transiter master theorem states that if $a\to b$ is a right tight function restriction expression for a subgroup property then the left transiter of that subgroup property is $b\to b$. Thus, the left transiter of normality is:

Automorphism $\to$ Automorphism

which is precisely the subgroup property of being characteristic.