Normal subgroup contained in the hypercenter is amalgam-characteristic

Statement
Suppose $$H$$ is a normal subgroup of $$G$$ contained in the fact about::hypercenter of $$G$$. In other words, $$H$$ is contained in some member of the transfinite upper central series of $$G$$. Then, $$H$$ is an amalgam-characteristic subgroup of $$G$$: $$H$$ is characteristic in the external amalgamated free product $$K := G *_H G$$.

Similar facts

 * Central implies amalgam-characteristic
 * Finite normal implies amalgam-characteristic
 * Periodic normal implies amalgam-characteristic

Opposite facts

 * Normal not implies amalgam-characteristic
 * Characteristic not implies amalgam-characteristic

Applications

 * Normal subgroup contained in hypercenter is potentially characteristic
 * Nilpotent implies every normal subgroup is potentially characteristic

Facts used

 * 1) uses::Quotient of amalgamated free product by amalgamated normal subgroup equals free product of quotient groups
 * 2) uses::Free product of nontrivial groups is centerless
 * 3) uses::Hypercenter is characteristic

Proof
Given: A group $$G$$, a subgroup $$H$$ contained in the hypercenter of $$G$$. $$L := G *_H G$$.

To prove: $$H$$ is characteristic in $$L$$.

Proof:


 * 1) By fact (1), $$K/H \cong G/H * G/H$$.
 * 2) $$K/H$$ is centerless: If $$H$$ is proper in $$G$$, this follows from fact (2). If $$H = G$$, then $$K/H$$ is trivial, hence centerless.
 * 3) $$H$$ is in the hypercenter of $$K$$: This is because $$H$$ is in the hypercenter of each of the factors.
 * 4) $$H$$ equals the hypercenter of $$K$$: If $$g$$ is in the hypercenter of $$K$$, the image of $$g$$ via the quotient map $$K \to K/H$$ is in the hypercenter of $$K/H$$. However, since $$K/H$$ is centerless, we get that the image of $$g$$ is trivial, so $$g \in H$$. Thus, $$H$$ is the hypercenter.
 * 5) $$H$$ is characteristic in $$K$$: This follows from the previous step and fact (3).