The probability that an element of a non-abelian group fixes a set and its applications in graph theory

The commutativity degree, defined as the probability that two randomly selected elements of a group commute, plays a very important role in determining the abelianness of a group. In this research, the commutativity degree is extended by finding the probability that a group element fixes a set. This probability is computed under two group actions on the set namely, the conjugate action and the regular action. The set under study consists of all commuting elements of order two of metacyclic 2-groups and dihedral groups of even order. The probabilities found turned out to depend on the cardinality of the set. The results which were obtained from the probability are then linked to graph theory, more precisely to orbit graph and generalized conjugacy class graph. It is found that the orbit graph and the generalized conjugacy class graph consist of complete graphs, empty graphs or null graphs. Moreover, some graph properties including the chromatic number, clique number, dominating number and independent number are found. In addition, the necessary condition for the orbit graph and generalized conjugacy class graph to be a null graph is examined. Furthermore, two new graphs are introduced, namely the generalized commuting graph and the generalized non-commuting graph. The generalized commuting graph of all groups in the scope of this research turns out to be a union of complete graphs or null graphs, while the generalized non-commuting graph consists of regular graphs, empty graphs or null graphs

The conjugation degree on a set of metacyclic 3-groups

Research on commutativity degree has been done by many authors since 1965. The commutativity degree is defined as the probability that two randomly selected elements in a group commute. In this research, an extension of the commutativity degree called the probability that an element of a group fixes a set Ω is explored. The group G in our scope is metacyclic 3-group and the set Ω consists of a pair of distinct commuting elements in the group G in which their orders satisfy a certain condition. Meanwhile, the group action used in this research is conjugation. The probability that an element of G fixes a set Ω, defined as the conjugation degree on a set is computed using the number of conjugacy classes. The result turns out to be 7/8 or 1, depending on the orbit and the order of Ω

On the generalized commuting and non-commuting graphs for metacyclic 3-groups

Let be a metacyclic 3-group and let be a non-empty subset of such that . The generalized commuting and non-commuting graphs of a group is denoted by and respectively. The vertex set of the generalized commuting and non-commuting graphs are the non-central elements in the set such that where Two vertices in are joined by an edge if they commute, meanwhile, the vertices in are joined by an edge if they do not commute

The probability that an element of metacyclic 2-groups of positive type fixes a set

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

The probability that a group element fixes a set underregular action formetacyclic 2-groups of negative type

In this paper, let be a metacyclic 2-group of negative type of class two and ofclass at least three. Let O be the set of all subsets of all commuting elements of size two in the form of a,b, where a and b commute and |a|= |b|= 2.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, the probability that an element of fixes the set O under regular action is computed

The conjugation degree on a set of metacyclic 3-groups

Research on commutativity degree has been done by many authors since 1965. The commutativity degree is defined as the probability that two randomly selected elements in a group commute. In this research, an extension of the commutativity degree called the probability that an element of a group fixes a set Ωis explored. The group G in our scope is metacyclic 3-group and the set Ω consists of a pair of distinct commuting elements in the group G in which their orders satisfy a certain condition. Meanwhile, the group action used in this research is conjugation. The probability that an element of G fixes a set Ω, defined as the conjugation degree on a set is computed using the number of conjugacy classes. The result turns out to be 7/8 or 1, depending on the orbit and the order of Ω