Parameterization and Graphical Representation of Certain Groups

Abstract

The objective of this research study is to investigate certain actions of the group 􀀀 de ned by linear-fractional transformations of the form r : z ! z 􀀀 1 z and s : z ! 􀀀1 2(z + 1) , satisfying the relations < r; s : r3 = s4 = 1 >. These actions can be on in nite elds (projective lines over real and imaginary quadratic elds) or on nite elds PL(Fq) for prime q. It has been shown that coset diagram for the actions of 􀀀 = G 3;4(2;Z) satisfying the relations < r; s : r3 = s4 = t2 = (rs)2 = (rt)2 = (st)2 = 1 > is connected and transitive on rational projective line. Using this, it is proved that for the group 􀀀, < r; s : r3 = s4 = 1 > are de ning relations. It is also found out that if is any real quadratic irrational number then a unique closed path could be formed by ambiguous numbers in the orbit 􀀀 . Actions of the group 􀀀 on PL(Fq) have been parameterized. It means that for a prime q and 2 Fq, a coset diagram D( ; q) represents each conjugacy class of actions of 􀀀 on PL(Fq) where q is a prime number. ln particular, each conjugacy class for actions of in nite triangle groups (3; 4; k) on PL(Fq) is associated with a coset diagram D( ; q).