J. Basic. Appl. Sci. Res., 3(10)369-380, 2013 © 2013, TextRoad Publication ISSN 2090-4304 Journal of Basic and Applied Scientific Research www.textroad.com * Corresponding Author: N. H. Sarmin, Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor, Malaysia. Email: nhs@utm.my. The Probability That an Element of a Group Fixes a Set and Its Graph Related to Conjugacy Classes S. M. S. Omer 1 , N. H. Sarmin 2,* and A. Erfanian 3 1 Department of Mathematics, Faculty of Science, Benghazi University, Libya 1,2 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia81310 UTM Johor Bahru, Johor, Malaysia 3 Department of Mathematics and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159, 91775, Mashhad, Iran Received: July 27 2013 Accepted: September 5 2013 ABSTRACT Let G be a metacyclic 2-group. The commutativity degree is the probability that two random elements commute in G , denoted as ( ) PG .This concept is used to determine the abelianness of a group and there are three ways to find this probability. The probability can be obtained by finding the multiplication table, conjugacy classes and recently using centralizers. In this paper, we use the conjugacy classes as a way to compute our results. The concept of commutativity degree has been generalized by many authors; one of these generalizations is the probability that a group element fixes a set. Thus, the main objective of this paper is to find the probability of an element of a group G fixes  where  is a set consisting of (, ) ab , where a and b are commuting elements in G of size two and the group G acts on the set of all subset of  by conjugation. The results that are obtained from the probability can be conducted with graph theory more precisely graph related to conjugacy classes. Hence, our second objective is to find the graph related to conjugacy class for the mentioned probability. The work in this article is done for all metacyclic 2-groups of nilpotency class two and class at least three of negative type. KEYWORDS: Commutativity degree, metacyclic 2-group, graph, conjugacy classes, group action. INTRODUCTION Throughout this paper, G denotes a non-abelian metacyclic 2-group. The probability that two randomly selected elements commute in G is called the commutativity degree of G , denoted by ( ) PG . The probability is defined as follows:     2 , ( ) xy G G xy yx PG G     The above probability is less than or equal to 5/8 for finite non-abelian groups [1,2]. Gustafson [1] and MacHale [2] showed that this probability can be computed using conjugacy classes. Numerous researches have been done on the commutativity degree and many results have been achieved. As mentioned, this probability has been generalized by lots of researches and these generalizations have also been extended by many authors. A part from that, we use one of those generalizations which is the probability that a group element fixes a set that was firstly introduced by Omer et al. [3]. This probability can be obtained by calculated the conjugacy classes under some group action on a set. In the following context, we state some basic concepts that are needed in this paper. These basic concepts can be found in one of the references [4, 5]. Definition1.1[4]A group G is called a metacyclic if it has a cyclic normal subgroup K such that the quotient group G H is also cyclic. 369