The probability that an element of a non-abelian group fixes a set and its applications in graph theory

Abstract

The commutativity degree, defined as the probability that two randomly selected elements of a group commute, plays a very important role in determining the abelianness of a group. In this research, the commutativity degree is extended by finding the probability that a group element fixes a set. This probability is computed under two group actions on the set namely, the conjugate action and the regular action. The set under study consists of all commuting elements of order two of metacyclic 2-groups and dihedral groups of even order. The probabilities found turned out to depend on the cardinality of the set. The results which were obtained from the probability are then linked to graph theory, more precisely to orbit graph and generalized conjugacy class graph. It is found that the orbit graph and the generalized conjugacy class graph consist of complete graphs, empty graphs or null graphs. Moreover, some graph properties including the chromatic number, clique number, dominating number and independent number are found. In addition, the necessary condition for the orbit graph and generalized conjugacy class graph to be a null graph is examined. Furthermore, two new graphs are introduced, namely the generalized commuting graph and the generalized non-commuting graph. The generalized commuting graph of all groups in the scope of this research turns out to be a union of complete graphs or null graphs, while the generalized non-commuting graph consists of regular graphs, empty graphs or null graphs.