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.

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

“…The probability was found for some finite non-Abelian groups including metacyclic 2-groups [8], symmetric groups and alternating groups [9]. Recently, Mustafa et al [4] extended the work in [3] by restricting the order of the set Ω. The following theorem illustrates their work.…”

“…In addition, Mustafa et al computed the probability that a group element fixes a set for metacyclic 2-groups of positive type of nilpotency class at least three [11]. In this paper, the results obtained from the probability that a group element fixes a set are then applied to graph theory.…”

“…The work in [4] has then been extended by finding the probability for some finite non-abelian groups such as the metacyclic 2-groups of negative type of nilpotency class two and class at least three [10]. In addition, Mustafa et al computed the probability that a group element fixes a set for metacyclic 2-groups of positive type of nilpotency class at least three [11].…”

“…Theorem 2.1 [4] Let G be a finite group. Let S be a set of elements of G of size two in the form of (a, b) where a and b commute, |a| = |b| = 2.…”

“…Omer et al [3] have extended the commutativity degree by defining the probability that a group element fixes a set. Extension has been done on the work in [3] by Mustafa et al [4], where the size of the fixed set was restricted.…”