Cyclic normal implies potentially verbal in finite

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Cyclic normal subgroup (?)) must also satisfy the second subgroup property (i.e., Potentially verbal subgroup (?)). In other words, every cyclic normal subgroup of finite group is a potentially verbal subgroup of finite group.
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Statement

Statement with symbols

Suppose $K$ is a finite group and $H$ is a cyclic normal subgroup of $K$. In other words, $H$ is a normal subgroup of $K$ that is also a cyclic group. Then, $H$ is a potentially verbal subgroup of $G$: there exists a group $G$ containing $K$ such that $H$ is a verbal subgroup of $G$. In fact, we can choose $G$ to itself be a finite group.

Related facts

• Cyclic normal implies finite-pi-potentially verbal in finite
• Homocyclic normal implies finite-pi-potentially fully invariant in finite
• Central implies potentially verbal in finite
• Homocyclic normal implies potentially fully invariant in finite
• Central implies finite-pi-potentially verbal in finite
• Central implies finite-pi-potentially characteristic in finite

Facts used

1. Cyclic normal implies finite-pi-potentially verbal in finite

Proof

The proof follows directly from fact (1), which is a somewhat stronger formulation.