Alternating groups as a quotient of $${{\varvec{PSL}}}\left( \mathbf{2}, \pmb {\mathbb {Z}} \left[ {{\varvec{i}}}\right] \right) $$ PSL 2 , Z i

Abstract

In this study, we developed an algorithm to find the homomorphisms of the Picard group $$\textit{PSL}(2,Z[i])$$ into a finite group G. This algorithm is helpful to find a homomorphism (if it is possible) of the Picard group to any finite group of order less than 15! because of the limitations of the GAP and computer memory. Therefore, we obtain only five alternating groups $$ A_{n}$$ , where $$n=5,6,9,13$$ and 14 are quotients of the Picard group. In order to extend the degree of the alternating groups, we use coset diagrams as a tool. In the end, we prove our main result with the help of three diagrams which are used as building blocks and prove that, for $$n\equiv 1,5,6(\mathrm { mod}\, 8)$$ , all but finitely many alternating groups $$A_{n}$$ can be obtained as quotients of the Picard group $$\textit{PSL}(2,Z[i])$$ . A code in Groups Algorithms Programming (GAP) is developed to perform the calculation.