Applications of Graphs Related to the Probability that an Element of Finite Metacyclic 2-Group Fixes a Set

In this paper, G denotes a metacyclic 2-group of positive type of nilpotency class at least three and Ω is the set of all subsets of commuting elements of G of size two in the form of (a, b), where a and b commute and lcm(|a|, |b|) = 2. The probability that a group element of G fixes a set is one of the generalizations of the commutativity degree that has been recently introduced. In this paper, the probability that an element of fixes a set for metacyclic 2-groups of positive type of nilpotency class at least three is computed. The results obtained are then applied to graph theory, more precisely to the orbit graph and generalized conjugacy class graph.