Characteristic of normal implies normal

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties , to another known subgroup property
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Statement

Property-theoretic statement

Characteristic * Normal $\leq$ Normal

Verbal statement

Every characteristic subgroup of a normal subgroup is normal.

Symbolic statement

Let $H \leq K \leq G$ such that $H$ is characteristic in $K$ and $K$ is normal in $G$ , then $H$ is normal in $G$ .

Related facts

Left transiter of normal is characteristic: Characteristicity is the weakest, or most general property, for which the above statement is true. This is made precise in the statement that characteristicity is the left transiter for normality.

Proof

Hands-on proof

Given groups $H \leq K \leq G$ such that $H$ is characteristic in $K$ and $K$ is normal in $G$ . We need to show that for any $g \in G$ , the map $c_{g} : x \mapsto g x g^{- 1}$ takes $H$ to within itself.

First, notice that since $K ◃ G$ , $c_{g} (x) \in K$ for every $x \in K$ . Thus, $c_{g}$ restricts to a function from $K$ to $K$ . Since this function arises by restricting an automorphism of $G$ , it is an endomorphism of $K$ .

Further, since $c_{g^{- 1}} \circ c_{g}$ is the identity map, and $K$ is invariant under both, the restriction of $c_{g}$ to $H$ is actually an invertible endomorphism, viz an automorphism. Call this automorphism $σ$ .

Since $H$ is characteristic in $K$ , $σ$ takes $H$ to within itself. But since $σ$ is the restriction of $c_{g}$ to $K$ in the first place, we conclude that $c_{g}$ in fact takes $H$ to itself.

Using the function restriction formalism

In terms of the function restriction formalism:

The following is a restriction formal expression for the subgroup property of normality:

Inner automorphism $\to$ Automorphism

The following is a restriction formal expression for the subgroup property of characteristicity:

Automorphism $\to$ Automorphism

We now use the composition rule for function restriction to observe that the composition of characteristic and normal implies the property:

Inner automorphism $\to$ Automorphism

Which is again the subgroup property of normality.