On the probability that a group element fixes a set and its generalized conjugacy class graph

Abstract

Let G be a metacyclic 2-group. The probability that two random elements commute in G is the quotient of the number of commuting elements by the square of the order of G. This concept has been generalized and extended by several authors. One of these extensions is the probability that an element of a group fixes a set, where the set consists of all subsets of commuting elements of G of size two that are in the form (a,b), where a and b commute and lcm(|a|, |b|) = 2. In this paper, the probability that a group element fixes a set is found for metacyclic 2-groups of negative type of nilpotency class at least two. The results obtained on the size of the orbits are then applied to graph theory, more precisely to generalized conjugacy class graph.