• Awais Yousaf

      Articles written in Proceedings – Mathematical Sciences

    • Diagrams for Certain Quotients of $PSL(2, \mathbb{Z}[i])$

      Qaiser Mushtaq Awais Yousaf

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      Actions of the Picard group $PSL(2,\mathbb{Z}[i])$ on $PL(F_p)$, where $p\equiv 1(\mathrm{mod} 4)$, are investigated through diagrams. Each diagram is composed of fragments of three types. A technique is developed to count the number of fragments which frequently occur in the diagrams for the action of the Picard group on $PL(F_p)$. The conditions of existence of fixed points of the transformations are evolved. It is further proved that the action of the Picard group on $PL(F_p)$ is transitive. A code in Mathematica is developed to perform the calculation.

    • Alternating groups as a quotient of $PSL (2,\mathbb{Z}[i])$


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      In this study, we developed an algorithm to find the homomorphisms of the Picard group $PSL(2, Z[i])$ 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(mod 8), all but finitely many alternating groups $A_n$ can be obtained as quotients of the Picard group $PSL(2, Z[i])$. A code in Groups Algorithms Programming (GAP) is developed to perform the calculation.

    • On contraction of vertices of the circuits in coset diagrams for $PSL(2,\mathbb{Z})$


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      Coset diagrams for the action of $PSL(2,\mathbb{Z})$ on real quadratic irrational numbers are infinite graphs. These graphs are composed of circuits. When modular group acts on projective line over the finite field $F_{q}$ , denoted by $PL(F_{q})$, vertices of the circuits in these infinite graphs are contracted and ultimately a finite coset diagram emerges. Thus the coset diagrams for $PL(F_{q})$ is composed of homomorphic images of the circuits in infinite coset diagrams. In this paper, we consider a circuit in which one vertex is fixed by $(xy)^{m_{1}} (xy^{−1)m_{2}}$, that is, $(m_{1},m_{2})$. Let $\alpha$ be the homomorphic image of $(m_{1},m_{2})$ obtained by contracting a pair of vertices $v$, $u$ of $(m_{1},m_{2})$. If we change the pair of vertices and contract them, it is not necessary that we get a homomorphic image different from $\alpha$. In this paper, we answer the question: how many distinct homomorphic images are obtained, if we contract all the pairs of vertices of $(m_{1},m_{2})$?We also mention those pairs of vertices, which are ‘important’. There is no need to contract the pairs, which are not mentioned as ‘important’. Because, if we contract those, we obtain a homomorphic image, which we have already obtained by contracting ‘important’ pairs.

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