Finite normal implies potentially characteristic

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 $K$.

Related facts

Stronger 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.
• NPC theorem: This states that any normal subgroup is potentially characteristic.

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).