Cyclicity is quotient-closed

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

If $G$ is a cyclic group and $H$ is a normal subgroup of $G$, the quotient group $G/H$ is also a cyclic group.

Proof

Suppose $G=\langle g\rangle$. Then the elements of $G/H$ are the cosets $g^{i}H$. $g^{i}H=(gH)^{i}$, so $G/H=\langle gH\rangle$.