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 $\le$ Normal

Here, $*$ denotes the composition operator.

Verbal statement

Every characteristic subgroup of a normal subgroup is normal.

Symbolic statement

Let $H\le K\le G$ 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 $H\le K\le G$ 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 $c_g:x\mapsto gx{g}^{-1}$ 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 $\sigma$.

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.

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info}, Page 135
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info}, Page 28, Characteristic and Fully invariant subgroups, 1.5.6(iii)
• Topics in Algebra by I. N. Herstein^{More info}, Page 70, Problem 9

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