Some applications of metacyclic 2-groups of negative type

Abstract

The probability that two random elements commute in a finite group G is the quotient of the number of commuting elements and |G|2. Consider a set S consisting 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. The probability that a group element fixes S is the number of orbits under the group action on S divided by |S|. In this paper, the probability that a group element fixes a set S under regular action is found for metacyclic 2-groups of negative type of nilpotency class two and of class at least three. The results obtained from the sizes of the orbits are then applied to the generalized conjugacy class graph.