Finite normal implies amalgam-characteristic

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., finite normal subgroup) must also satisfy the second subgroup property (i.e., amalgam-characteristic subgroup)
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Statement

Verbal statement

Any finite normal subgroup (i.e., a normal subgroup that is finite as a group) is an amalgam-characteristic subgroup.

Statement with symbols

Suppose $H$ is a finite normal subgroup of a group $G$. Then, $H$ is a characteristic subgroup inside the amalgam $L:=G{*}_{H}G$.

Related facts

• Central implies amalgam-characteristic
• Normal not implies amalgam-characteristic

Applications

• Finite normal implies potentially characteristic
• Finite implies every normal subgroup is potentially characteristic

Proof

Given: A group $G$, a finite normal subgroup $H$ of $G$. $L:=G{*}_{H}G$.

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

Proof:

1. Any element outside $H$ has infinitely many conjugate elements, i.e., its conjugacy class is infinite in size.
2. $H$ is the unique largest finite normal subgroup in $L$.
3. $H$ is characteristic in $L$.