The Probability that an Element of Metacyclic 2-Groups of Positive Type Fixes a Set

Abstract

In this paper, is a metacyclic 2-group of positive type of nilpotency of class at least three. Let  be the set of all subsets of all commuting elements of of size two in the form of where and commute and each of order two. The probability that an element of a group fixes a set is considered as one of the extensions of the commutativity degree that can be obtained by some group actions on a set. In this paper, we compute the probability that an element of fixes a set in which acts on a set, by conjugation.