Subgroups and quotients of a cyclic group

A nontrivial subgroup $H$ of $⟨g⟩$ is generated by $g^d$, where $d$ is the least positive exponent with $g^d\in H$. Divide any exponent $k$ with $g^k\in H$ by $d$; its remainder also gives an element in $H$ and must be zero. A quotient group by a normal subgroup is generated by the coset of $g$, because every coset is a power of that coset. The trivial subgroup and quotient are also cyclic.