Subgroups and quotients of a cyclic group
= Subgroups and quotients of a cyclic group
A nontrivial <subgroup> $H$ of $\langle g\rangle$ 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.