I'm wondering how can we know a quotient group forms a cyclic group.
Suppose we have a quotient $(\mathbb{Z}/8\mathbb{Z})^*/\langle 3\rangle=\{\{1,3\},\{5,7\}\}$ where $(\mathbb{Z}/8\mathbb{Z})^*=\{1,3,5,7\}$ is a multiplicative group.
This forms a cyclic group since a set of representative composed of each representative for each element is a cyclic group, i.e., $\{1,3\}=\langle 3\rangle$
How can we determine representatives for a quotient group?