Cyclic-quotient characteristic implies upward-closed characteristic

Statement
Suppose $$G$$ is a group and $$H$$ is a cyclic-quotient characteristic subgroup of $$G$$. In other words, $$H$$ is a fact about::characteristic subgroup of $$G$$ that is also a fact about::cyclic-quotient subgroup of $$G$$, i.e., the quotient group $$G/H$$ is a cyclic group.

Then, $$H$$ is an upward-closed characteristic subgroup of $$G$$: every subgroup $$K$$ of $$G$$ containing $$H$$ is a characteristic subgroup of $$G$$.

Related facts

 * SQ-dual::Cyclic characteristic implies hereditarily characteristic
 * Equivalence of definitions of upward-closed normal subgroup

Facts used

 * 1) uses::Cyclic implies every subgroup is characteristic
 * 2) uses::Characteristicity is quotient-transitive: If $$A \le B \le C$$ with $$A$$ characteristic in $$C$$ and $$B/A$$ characteristic in $$C/A$$, then $$B$$ is characteristic in $$C$$.

Proof
Given: A group $$G$$, a characteristic subgroup $$H$$ of $$G$$ such that $$G/H$$ is cyclic, a subgroup $$K$$ of $$G$$ containing $$H$$.

To prove: $$K$$ is characteristic in $$G$$.

Proof:


 * 1) $$K/H$$ is characteristic in $$G/H$$: This follows from fact (1), and the given datum that $$G/H$$ is cyclic.
 * 2) $$K$$ is characteristic in $$G$$: This follows from the previous step, the given datum that $$H$$ is characteristic in $$G$$, and fact (2).