Finite normal implies potentially 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., potentially characteristic subgroup)
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Statement

Suppose $G$ is a group and $H$ is a finite normal subgroup of $G$: $H$ is a normal subgroup of $G$ that is finite as a group. Then, there exists a group $K$ containing $G$ such that $H$ is characteristic in $K$.

Definitions used

Potentially characteristic subgroup

Further information: Potentially characteristic subgroup

A subgroup $H$ of a group $G$ is termed a potentially characteristic subgroup if there exists a group $K$ containing $G$ such that $H$ is a characteristic subgroup of $K$.

Related facts

• Finite NPC theorem: This states that a normal subgroup of a finite group can be realized as a characteristic subgroup in some finite group containing it.

Facts used

1. Finite normal implies amalgam-characteristic
2. Amalgam-characteristic implies potentially characteristic

Proof

The proof follows directly by piecing together facts (1) and (2).