Fully invariant of strictly characteristic implies strictly characteristic

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Fully invariant subgroup (?) and Strictly characteristic subgroup (?)), to another known subgroup property (i.e., Strictly characteristic subgroup (?))
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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., fully invariant subgroup) must also satisfy the second subgroup property (i.e., left-transitively strictly characteristic subgroup)
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Statement

Property-theoretic statement

Fully invariant * Strictly characteristic $\le$ Strictly characteristic

Verbal statement

Every fully invariant subgroup of a strictly characteristic subgroup is strictly characteristic.

Symbolic statement

Let $H\le K\le G$ such that $H$ is fully invariant in $K$ and $K$ is strictly characteristic in $G$, then $H$ is strictly characteristic in $G$.

Proof

Hands-on proof

Given groups $H\le K\le G$ such that $H$ is fully invariant in $K$ and $K$ is strictly characteristic in $G$. We need to show that for any surjective endomorphism $\sigma$ of $G$, $\sigma$ takes $H$ to within itself.

First, notice that since $K$ is strictly characteristic in $G$, $\sigma (x)\in K$ for every $x\in K$. Thus, $\sigma$ restricts to a function from $K$ to $K$. Since this function arises by restricting an endomorphism of $G$, it is an endomorphism of $K$.

Since $H$ is fully invariant in $K$, $\sigma {|}_K$ takes $H$ to within itself. But since $\sigma {|}_K$ is the restriction of $\sigma$ to $K$ in the first place, we conclude that $\sigma$ in fact takes $H$ to itself.

Using the function restriction formalism

In terms of the function restriction formalism:

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

Surjective endomorphism $\to$ Endomorphism

• The following is a function restriction expression for the subgroup property of full invariance:

Endomorphism $\to$ Endomorphism

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

Surjective endomorphism $\to$ Endomorphism

Which is again the subgroup property of strict characteristicity.