Order product prime graph and its variations of some finite groups

Abstract

The study of groups from geometric viewpoint has recently become one of the focus of researches in group theory, which started with the Cayley graph. Later, the study grew through the years, leading to the definition of many graphs of groups and investigation of graphical properties of finite groups. This development exists due to the fact that groups can be profitably studied as geometric objects in their own right, since the geometry exists both in the group itself and in the spaces it acts on. This study basically shows how groups and spaces interact together, which helps in understanding the symmetries of much more complicated objects. In this thesis, the order product prime graph of finite groups is defined as the graph whose vertices are the elements of the groups, and any two vertices are adjacent if and only if the product of their orders is a prime power. Moreover, another graph which is commuting order product prime graph of finite groups is defined as the graph whose vertices are the elements of finite groups, and any two vertices are adjacent if and only if they commute and the product of their order is a prime power. Furthermore, these definitions are extended to the order prime permutability graph of subgroups of finite groups as the graph whose vertices are the proper subgroups of finite groups, and any two vertices are adjacent if and only if they permute and the product of their order is a prime power. Also the order prime permutability graph of cyclic subgroups of finite groups is defined as the graph whose vertices are the proper subgroups of finite groups, and any two vertices are adjacent if and only if they are permuting cyclic subgroups and the product of their orders is a prime power. The order product prime graph is connected, complete and regular on all quasi-dihedral groups, cyclic groups of prime power order and generalized quaternion groups, Q4n, where n is even prime power. On dihedral groups, the graph is connected only if the degree is prime power, but complete and regular if the degree is even prime power. The commuting order product prime graph is connected, complete and regular on cyclic groups of prime power order and connected on quasi-dihedral groups, dihedral groups of prime power degree and generalized quaternion groups, Q4n, where n is even prime power. Next is the order prime permutability graph of subgroups, which is connected, complete and regular on cyclic groups of prime power order and connected on quasi-dihedral groups, dihedral groups, Dn and generalized quaternion groups, Q4m, where m is even prime power or just prime. Finally, the order prime permutability graph of cyclic subgroups, is connected, complete and regular on cyclic groups of prime power order and connected on dihedral groups of prime degree and generalized quaternion groups, Q4p. The properties of the graphs are used in obtaining their invariants on cyclic groups, dihedral groups, generalized quaternion groups and quasi-dihedral groups, which include the clique number, independence number, domination number, girth, diameter, vertex chromatic number, edge chromatic number and some other recently introduced chromatic numbers, which are the dominated chromatic number and locating chromatic number. Moreover, the general presentations of the graphs on the above groups are used in exploring the number of perfect codes of the graphs, which has also been recently introduced on graphs of groups.