# Cyclic normal implies potentially verbal in finite

From Groupprops

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 is a finite group and is a cyclic normal subgroup of . In other words, is a normal subgroup of that is also a cyclic group. Then, is a potentially verbal subgroup of : there exists a group containing such that is a verbal subgroup of . In fact, we can choose 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

## Proof

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