Cyclic normal implies hereditarily normal

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., cyclic normal subgroup) must also satisfy the second subgroup property (i.e., hereditarily normal subgroup)
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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
View other applications of characteristic of normal implies normal OR Read a survey article on applying characteristic of normal implies normal

Statement

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 $G$.

References

Textbook references

• Topics in Algebra by I. N. Herstein, ^{More info}, Page 53, Problem 13