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

Here, $*$ denotes the composition operator.

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.
• Automorph-permutable of normal implies conjugate-permutable: This statement has many corollaries; for instance, 2-subnormal implies conjugate-permutable

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 $c_{g}:x\mapsto gxg^{-1}$ takes $H$ to within itself.

First, notice that since $K\triangleleft G$, $c_{g}(x)\in K$ 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 $\sigma$.

Since $H$ is characteristic in $K$, $\sigma$ takes $H$ to within itself. But since $\sigma$ 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.

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