Cyclic normal implies hereditarily normal

Verbal statement
Any subgroup of a cyclic normal subgroup is normal. (Also, any subgroup of a cyclic normal subgroup is cyclic, so in fact, subgroups of cyclic normal subgroups are cyclic normal).

Property-theoretic statement
The subgroup property of being a cyclic normal subgroup is stronger than the subgroup property of being a hereditarily normal subgroup. Equivalently, the property of being cyclic normal is a left-hereditary subgroup property.

Facts used

 * 1) Any subgroup of a cyclic group is a characteristic subgroup thereof
 * 2) A characteristic subgroup of a normal subgroup is normal

Proof
Given: A group $$G$$, a cyclic normal subgroup $$H$$, and a subgroup $$K$$ of $$H$$

To prove: $$K$$ is normal in $$G$$

Proof: By fact (1), and the given fact that $$H$$ is cyclic, $$K$$ is characteristic in $$H$$. By fact (2), and the given datum that $$H$$ is normal in $$G$$, we conclude that $$K$$ is normal in $$G$$.

Textbook references

 * , Page 53, Problem 13