Projective Equivalence for the Roots of Unity

Let ${\mu }_{\mathrm{\infty }}\subseteq \mathbb{C}$ be the collection of roots of unity and $$\mathcal{C}${n}:={(s{1},\cdots,s_{n})\in\mu_{\infty}^{n}:s_{i}\neq s_{j}$\text{ for any }$1\leq i<j\leq n}$\mathrm{.}Twoelements$(s_{1},\cdots,s_{n})$and$(t_{1},\cdots,t_{n})$of$$\mathcal{C}$_{n}$ are said to be projectively equivalent if there exists $\gamma \in \text{PGL}(2,\mathbb{C})$ such that $\gamma (s_i)=t_i$ for any $1\le i\le n$. In this article, we will give a complete classification for the projectively equivalent pairs. As a consequence, we will show that the maximal length for the nontrivial projectively equivalent pairs is $14$.