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

  • Qaiser Mushtaq
  • Awais Yousaf


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.


Bianchi group fragment orbits groups algorithms programming 

2010 Mathematics Subject Classification

Primary: 05C25 Secondary: 20G40 


  1. 1.
    Fine B, Algebraic theory of Bianchi groups (1989) (Marcel Dekker Inc., New York)zbMATHGoogle Scholar
  2. 2.
    Fine B and Frohman C, Some amalgam structures for Bianchi groups, Proc. Amer. Math. Soc. 102 (1988) 221–229MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Mushtaq Q and Razaq A, Equivalent pairs of words and points of connection, Sci. World J., 2014 (2014)Google Scholar
  4. 4.
    Mushtaq Q and Yousaf A, Diagrams for certain quotients of \( PSL(2,Z[i])\), Proc. Indian Acad. Sci. (Math. Sci.), 124 (3) (2014) 291–299MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ratcliffe J G, Foundations of hyperbolic manifolds, 149 (2006) (Springer)Google Scholar
  6. 6.
    Sansone G, I sottogrupppe del gruppo di Picard e due teoremi sui grupppi finiti analoghi at teorema del dyck, Rend Circ. Mat., Palermo, 46 (1923) 273, cited from: Fine B, Algebraic theory of Bianchi groups (1989) (New York: Marcel Dekker Inc.)Google Scholar
  7. 7.
    Serre J P, Le Probleme de groupes de congruence pour \( SL_{2}\), Ann. Math., 92 (1970) 489–657, cited from: Fine B, Algebraic theory of Bianchi groups (1989) (New York: Marcel Dekker Inc.)Google Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe Islamia University of BahawalpurBahawalpurPakistan

Personalised recommendations