LINEAR GROUPS AND THEIR ACTIONS ON CERTAIN FIELDS

Abstract

An extension of degree 2 of the field of rational numbers Q is called the quadratic fie ld. Since Q(,JYf':') = Q(..rn; ) if and only if n 1 = c2 n 2' where CEQ, therefore any quadratic field has the form Q(rn), where n is a square-free integer that is uniquely determined by the field. In what follows, n will always be taken to be this integer. When n > 0, Q(rn) is called a real, and when n < ° an imaginary, quadratic field. It is worthwhile to consider linear-fractional transformations x,y satisfying the relations x2 = ym = 1, with a view to studying an action of the group < x,y > on real quadratic fields. If y : z ~ az+db is to act on all real quadratic fields then a, b, c, d must be rational numbers and can cz+ be taken to be integers, so that (ad +d)b 2 is rational. But if y : z ~ az+db is of order m one must a - c cz+ have ~:_ : = 0) 2+0)-2+2, where OJ is a primitive m - th root of unity. Now OJ 2 + (0 -2 is rational, for a primitive m - th root OJ, only if m = 1,2,3,4 or 6. So these are the only possible orders of y . The group < x,y > is C2 (cyclic group of order 2) when m = 1. When m = 2 , it is an infinite dihedral group and does not give inspiring information while studying its action on the real quadratic irrational numbers. For m = 3, the group < x, y > is the modular group PSL(2,Z). A real quadratic irrational number a = a+;n is said to be totally positive if a and its algebraic conjugate a are both positive and said to be totally negative if both a and a are negative. A real quadratic irrational number a = a+;n is said to be ambiguous if both a and a are of opposite signs. It is known that the group G2,6 (2, Z) = < x,y : x2 = y6 = 1 > is generated by the linear . fractional transformations x and y, where (z)x = ~~ and (z)y = 3( ~1) are defined on the set of integers. If we let u = y , v = xyx then u, v can be considered as the linear fractional transformations defined by (z)u = 3(~1 ) and (z)v = 3;~1 . So the group G6,6(2,Z) = < u,v > is a subgroup of the group G2,6 (2, Z). That is, G6,6 (2, Z) = < u, v : u6 = v6 = 1 > is the group of linear fractional transformations of the form z ~ az+db , where a, b, e, d E Z and ad - be = 1 or 3. cz+ The linear-fractional transformation t:z ~ t inverts u and v, that is, t 2 = (ut) 2 = (vt) 2 = 1 and so extends the group G6,6 (2, Z) to G6,6 (2, Z). The extended group G6,6 (2, Z) has presentation < u, v,t : u6 = v6 = t2 = (ut)2 = (vt) 2 = 1 >. Triangle groups are represented by 6(l,m,n) = < x,y : Xl = ym = (xy)n = 1 >, where l,m,n are positive integers greater than or equal to one. It is well-known that 6(l,m,n) is isomorphic to.a subgroup of PSL(2,C). Let q be a prime. Then by the projective line over the finite field Fq, we mean Fq U {oo}. We denote it by PL(Fq ). The group G6,6(2,q) is then the group of linear fractional transformations of the form z ~ ::~, where a, b,e,dEFq and ad - be '* 0, while G6,6(2, q) is its subgroup consisting of all those linear fractional transformations of the form z ~ ::!, where a,b,c,d E Fq and ad - be is a non-zero square in Fq. This thesis comprises four chapters. The aim of chapter one is to provide background material for succeeding chapters. In chapter two, we show that for a given totally positive (negative) real quadratic irrational number there exists an alternating sequence of totally positive and totally negative numbers which terminate at an ambiguous number. The ambiguous numbers form a closed path in the 2 coset diagram of the orbit aG, where a E Q(Jn) and this is the only closed path in the diagram. We also show that the action of G6,6(2, Z) on the rational projective line is transitive and the coset diagram of this action is connected. Finally, we show that u6 = v6 = 1 are the defining relations for the group G6,6(2,Z). At the end, we show that the action of G6,6(2,Z) on Q(Jn) is intransitive. In chapter three, we study action of the group G6,6 (2, Z) on the imaginary quadratic field Q( Fn) by using coset diagrams. In this chapter we show that the subset {a+-;:" : a, a 2 ;n , C E Z,c '* O} of Q(Fn) is invariant under the action of G6,6 (2,Z) on Q(Fn) and the fixed points of the non identity elements of G6,6(2,Z) exists only when it acts on Q*(H) = {a+f3 : a, a~; 3 ,c E Z,c '* O}. Also we show that the total number of orbits under the action of G6,6(2,Z) on the set Q*(Fn ) = {a+;; : a, a;;n ,c E Z,c '* O}, when n '* 3, are 2d(k) for n = 3k, k E Z and 4[d(k + 1) + d(k + 2) - 2] for n = 3k + 2, k E z, where den) is the arithmetic function. At the end, we show that the action of G6,6(2,Z) on Q(Fn) is intransitive. In chapter four, we parameterize the conjugacy classes of non-degenerate homomorphisms which represent actions of .0.(6,6,k) = < u, v : u6 = v6 = (uv)k = 1 > on the projective line over Fq , PL(Fq ), where q == ±l(modk). Also, for various values of k, we find conditions for the existence of coset diagrams depicting the permutation actions of .0.(6,6,k) on PL(Fq ). The conditions are polynomials with integer coefficients and the diagrams are such that every vertex in them is fixed by (Z:l""vY. In this way , we get .0.(6,6,k) as permutation groups on PL(Fq ). Also, we parameterize actions of G6,6(2,Z) on PL(Fq ) by the elements of Fq . We prove that the conjugacy classes of non-degenerate homomorphisms (J are in one-to-one 3 correspondence with the conjugacy classes of non-trivial elements of F q , under a correspondence which assigns to the homomorphism a the class containing (uv)a. Of course, this will mean that we can actually parameterize the actions of G6. 6 (2,Z) on PL(Fq ) by the elements of Fq • We develop a useful mechanism by which we can construct a unique coset diagram for each conjugacy class of these non-degenerate homomorphisms which depict the actions ofG6. 6 (2,Z) onPL(Fq ). A paper containing results from chapter two has been published [M.Aslam and Q.Mushtaq, Closed paths in the coset diagrams for < y, ( : y6 = (6 = 1 > acting on real quadratic fields, Ars Comb., 71(2004) 267-288.]. Another paper containing some results from this chapter has been accepted in the International Journal of Mathematics, Game Theory and Algebra. A paper containing results from chapter three has been submitted in an international journal for publication. Two papers containing results from chapter four have also been submitted for publication in international journals.