Central implies amalgam-characteristic

Statement with symbols
Suppose $$H$$ is a central subgroup of a group $$G$$. Then, $$H$$ is a characteristic subgroup inside the amalgam $$K := G *_H G$$. In other words, $$H$$ is an amalgam-characteristic subgroup.

Central subgroup
A subgroup $$H$$ of a group $$G$$ is termed a central subgroup if every element of $$H$$ commutes with every element of $$G$$. Equivalently, $$H$$ must be contained in the center of $$G$$.

Amalgam-characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed an amalgam-characteristic subgroup if $$H$$ is a characteristic subgroup inside the amalgam $$L := G *_H G$$.

Similar facts

 * Normal subgroup contained in hypercenter is 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

 * Central implies potentially characteristic
 * Abelian implies every 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::Center is characteristic

Proof
Given: A group $$G$$, a central subgroup $$H$$. $$L := G *_H G$$.

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 center of $$K$$: This is because $$H$$ is in the center of each of the factors.
 * 4) $$H$$ equals the center of $$K$$: If $$g$$ is in the center of $$K$$, the image of $$g$$ via the quotient map $$K \to K/H$$ is in the center 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 center.
 * 5) $$H$$ is characteristic in $$K$$: This follows from the previous step and fact (3).