Keywords (Extracted from title, table of contents and abstract of thesis)
coset diagrams, ambiguous numbers, ambiguous integers, ambiguous units, ambiguous primes, graphical representation, quadratic field 
Abstract Coset Diagrams are a graphical representation of the permutation action of the groups. Studying groups through their actions on different sets and algebraic structure has become a useful technique to know about the structure of the groups. The graphs have played a vital role in studying these actions. The main object of this work is to examine the actions of infinite groups G = < x,y; x2 = y3 =1>, H = < x, y; x2 = y4 = 1> and M = < x, y; x2 = y6 = 1> on real quadratic fields Q(âˆšn) and to find the subsets of Q(âˆšn) invariant under the action of each of these groups. Certain proper subsets of Q(âˆšn) invariant under the actions of each of these groups G, H and M, Have also been discussed in this dissertation. In this dissertation, a type of graphs, called coset diagrams, is employed to investigate the orbits of certain subsets Q*(âˆšp), Q//(âˆšp) and Q///(âˆšp) of Q(âˆšp), p a rational prime, under the actions of the groups G, H and M respectively. This dissertation is concerned with the determination of number of ambiguous numbers, ambiguous integers, ambiguous units and ambiguous primes in certain subsets Q*(âˆšn), Q//(âˆšn) and Q///(âˆšn) of Q(âˆšn) which are invariant under the action of the groups G, H an M respectively. One of the principal results of this dissertation is that we have determined, for each non square positive rational integer n, the actual number of ambiguous numbers in Q*/(âˆšn), as a function of n.
