# Difference between revisions of "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 must also satisfy the second subgroup property
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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 $G$.